Ambiguity - Mathematics.

[Wizard19]

DD Y3 had a question.

How would you answer the question, How many tens in 100?
How many tens in 427?

DD who is very able mathematically answered the latter as 42.
Expected answer was 2.

If that is the correct answer, then it follows that there are zero tens in 100!

How would you mark that as a teacher, as children work in mixed abilities and one answer shows understanding beyond the basic place value?

I agree, there are 42 10's in 427 but I suspect the question is just very badly worded and they are wanting them to partition 427, so 4 hundreds, 2 tens and 7 units. Very badly worded indeed.

Agreed that DD's answer was correct and more natural response for any child who has started to understand division - unless it was in a set of questions which clearly stated it was about place value? But even then I would expect the teacher to mark it in a "well done for correctly dividing by 10, although this was really a question on place value" type way.

I've seen that ambiguity before on here.

What I'd say to the child is it depends on what you are currently doing: If you're doing place value, the answer is 2. If you're doing division then the answer is 10.

Wizard - I don't actually see this as a division question - but a question about what the digits in these numbers represent. I think this is ambiguously written - but what they're actually looking for is that your child understands the meaning of the digit in the hundreds/ tens/ units column.

So in the number 100 - the digits are 1 hunded/ 0 tens/ 0 units

This is how we write 100 items in base 10 notation.

100 in base 2 where we only have 0 - 1 to work with (so computers) would be the correct notation for our number 4 in base 10. (more info on binary numbers here: www.mathsisfun.com/binary-number-system.html)

so the position of the digit represents how many of that type of number (hundreds/ tens/ units) you have.

So in the number 427

there are 4 hundreds
2 tens
7 units

although 42 tens is not correct - and thinking about this as dividing by ten is understandable - as a written number you can't have 42 in the tens column - so the 4 is carried to the next column along (the hundreds in this case) - meaning 4 hundreds/ 2 tens/ 7 units.

In the number 50,427 the digits would represent:

5 ten thousands
0 thousands
4 hundreds
2 tens
7 units

There are some fun place value games here on Woodlands Junior Maths Zone: resources.woodlands-junior.kent.sch.uk/maths/interactive/numbers.htm#Place - the bead numbers game (which is like an abacus) really helps you to visualise that each individual number 0 - 9 (a single digit) in a column represents how many of that type of number (thousands/hundreds/ tens/ units/ etc...) you have.

HTH

This may be a little young for your DD - but this song from Mr. R may help:

I'm guessing by the way you phrased the OP that the answer to how many 10s in 100 was 10 and was marked correctly. On which case 42 tens in 427 should also have been marked correct.

As said above they were almost certainly looking at place value but that should have been made clear either in the title of the work or in the question.

LittleMissGreen:

Wizard19 wrote:

How would you answer the question, How many tens in 427?

DD who is very able mathematically answered ... as 42.
Expected answer was 2
(expected answer - correct answer according to teacher)

In order for that answer to be correct - this isn't about how many times the number 10 can go into 100 or 427 - this has to be about the meaning of the position of the digit as written - which is termed 'place value'

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You can't have 42 tens (the tens column can only take the digits 0,1,2,3,4,5,6,7,8 and 9).

so in 427

you have 7 UNITS
2 TENS
4 HUNDREDS

So if you're thinking of a system where you only have the digits 0 - 9

then

zero = 0
one = 1
two = 2
three = 3
four = 4
five = 5
six = 6
seven = 7
eight = 8
nine = 9
and ten - can't have two numbers in the same column so is written as 10
(literally - I have one ten and no units)

eleven would be 11 - literally I have one ten and one unit.

That's why 100 in base two is our normal number 4.

base 2 only uses 0 and 1 (binary numbers - as in how computers work).

zero = 0
one = 1
two = 10 (you can't write 2 and you can't put another digit in the column)

three = 11
four = 100 (link to working with binary numbers in my post above).

--------

For the moment we're working in base 10 (our normal day to day number system) - so once we hit 9 we have to add a column (the tens column) and once we hit 99 we have to add another column (the hundreds column) and so on.

More info on place value here: www.bbc.co.uk/bitesize/ks2/maths/number/place_value_headings/read/1/

there's a game & a quiz there for more practice as well Wizard19 & anyone else that's interested.

HTH

PSBD I completely get that, but as she also gave the question about 100 which her daughter appeared to get right with the answer of 10 then there is an issue.

LittleMissGreen -

Wizard19 only posted the once - at the top.

She did not give her daughter's answer for the how many tens in 100 question - but gave 42 for the how many tens in 427 question.

she then said the 'expected answer' (by which I presume she means the 'correct answer according to the teacher') for the latter question (427) is 2 (2 tens in 427) and then asks

If that is the correct answer, then it follows that there are zero tens in 100!

Which I read (based on ! point) as suggesting that was a bit mad there can't be no tens in 100? can there?

So I was saying yes - there can be no tens in 100 if you're talking about place value? in this case there is 1 hundred/ zero tens/ zero units.

The statutory requirement for Year 4 maths pupils is:

recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones

p24 top: www.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf

It wasn't how I was taught - and it also caught me out the first time there was a question like this - but I strongly suspect given what Wizard19 posted about the correct answer for How many tens in 427? is 2 - that this is what the teacher was getting at.

One of my longstanding bug-a-bears is that most primary homework in maths does not come home with a worked example(s) or an explanation of terminology for parents. It really should.

To Clarify, this question was part of a test covering many topics.

Some above are automatically assuming place value, talking about the tens column etc.

Extract

2. Write the next number in this sequence 16 14 12 10 ?
3. Write the number 360 in figures. Write the number 204 in figures

4. How many tens are there in 427?

So in the above context, there is no context - thus me calling it ambiguous.

Wizard19:

Yep - looking at this - it's going through various pattern skills/ number 'logic' - so question 4 isn't a straightforward division question (427 divided by 10 - 42 remainder 7) but a place value question.

As you said the answer to How many tens in 427 was 2 (in the book/ from the teacher - not clear?)

I think the context - is this is a series of questions not about calculation skills (addition/ subtraction/ multiplication/ division) - but about interpretation of numbers: what the digits stand for (both that 427 is written four-hundred and twenty-seven and that it means 4 hundreds, 2 tens and 7 units (as if counted on an abacus). Looks like a sequence/ Meaning test.
----
2. Write the next number in this sequence 16 14 12 10 ?

My guess would be 8 (n-2 sequence)
---------

3. Write the number 360 in figures. Write the number 204 in figures

I get that this is asking to fully write out 360 in words - but I personally would relate 'figures' to 'numbers' not words.

Write the number three-hundred and sixty in figures - for asking someone to write it in digits - 360.

------

Maybe feedback to the teacher that you found what was being asked very confusing - and possibly poorly worded on occasion.

HTH

Even as a place value question there are 42 tens in 427. There are also 427 ones or units.

I would have marked it correct and then kicked myself for phrasing the question badly. It should have been something along the lines of 'what is the value of the digit '2'in 427?' or 'what digit is in the tens place in 427?'

This exact question gets posted at the same time every year
Must be in the national curriculum for Year 3 maths!!

I think the last one was in chat wasn't it and ended up in a huge bunfight between two posters that carried on over several threads.

We haven't had a BIDMAS question for a while. Must be due one about now.

Thanks Rafa.., Ali.., Sock..., De.. , Etc

For understanding and responding to query as posted.

Rafa:

the point is place value is also part of teaching BASE 10 (our numeric system) and how it works

You only have single digits in the columns

when you answer 42 - for How many TENS in 427 your are muddling

how many times can 10 go into 427

with

what digit appears in the TENS column.

How many UNITS in 427?
How many HUNDREDS 427?
How many MILLIONS 1,567,427?

These questions are always about what digit appears in that column - so what do the digits symbolize (because numbers are symbols we've adopted to describe numeric concepts - but there are other means of doing this - see Roman numerals or Egyptian numerals).

It's a subtle but important distinction getting at the meaning/ translation of numeric digits.

Wizard19:

Again - I know you disagree with my responses and want 42 to be the answer to how many tens in 427 - but the 'TENS' term is the name of the column - and the answer is 2 (as you have said in your OP).

I get that you rate being able to see that 10 can go into 427 42 times at a higher ability level than the knowledge that 2 is in the TENS column.

What I've tried to show you is that place value as a concept is on the new national curriculum for Year 4 and division by 10 is introduced from Year 2 - so technically speaking comprehending place value is seen as higher order ability.

I get that it is oddly worded and that you learned differently - I learned differently and as I said it caught me out as well - but maths as taught now doesn't use 'TENS columns' or 'TENS digits' as Rafa described in his re-worded question...Indeed I don't think any primary uses 'digits' as a term - mores the shame.

on a KS2 SATs test your child will be asked

How many hundreds in 1657. and it will mean What digit appears in the HUNDREDS column.

That was the point I was trying to make - I'm sorry if you don't feel it was a helpful answer to your question - but I was trying to explain why 2 was the 'expected answer' and why (as I presumed) 42 would be marked wrong.

Ooh Rafa I love a good BIDMAS thread, me!

We had this thread last year I remember I started getting confused at some point because it started getting clever!

Both dc say it 2 btw. One is year 6 and one an above average year 3.

Oh now year 6 ds says it could be 42 depending on what the teacher wants to know but that he didn't know that in year 3.

And yes rafa I remember it being a bit bun fighty too.

It isn't muddling those things though. How we write numbers with only one digit from 0-9 in a column is only a very small part of place value and base 10.

When teaching I got children to exchange in both directions using diennes e.g. 16 ones into 1 ten and 6 ones and 1 ten and 6 ones into 16 ones or 2 tens into 20 ones or 3 hundreds into 30 tens. Understanding this equivalence is key. It's the basis on which our number system works and underpins the concept of borrowing in the standard written calculation.

42 might not be the answer the teacher was looking for but it isn't actually wrong given the lack of context to the question. Most worksheets I've seen tend to avoid this wording precisely because of this confusion.

Rafa:

I think there is an important difference

place value is entirely about what the position & value of digits symbolises within numbers written out as numerals (in digits) - For new mathematicians it can be easy to see 22 as being two 2s or 4 - as DD1 did - so it's important the understand that the position of the digit conveys meaning and that each column has a value increasing by powers of 10 (like an abacus).

diennes - which I grant you requires an appreciation of place value - is really about borrowing from/ carrying over to the column to the left - and is therefore less about place value and more calculation skill - appreciating that in the problem 13 - 7 you do not have enough units in the UNITS column to successfully take 7 from 3, so will need to borrow (converting 1 ten into 10 units). It requires an appreciation of place value (knowing that the position conveys a value meaning) but is more functional - really about calculation skills.

by your logic I could argue diennes is really about all number bonds - knowing 13 - 7 is 6 and the converse 6 + 7 = 13 and you'll need to carry that ten.

I do apologize to be nitpicking - but

1. I think it is an important distinction

2. your suggestion that the answer '42' could also right (for the question how many tens in 427) - just muddles poor Wizard19 about why that teacher marked it wrong and was 'expecting' the number 2 for the answer.

I presume you are a maths teacher at primary level. I'm a little surprised you haven't been willing to review why a colleague would be saying '2' is the correct answer. (I get that Wizard19 using 'expected' rather than 'correct' was a conscious ploy to suggest that '2' couldn't be correct from how many tens in 427 - and indeed that 0 for how many tens in 100 'can't be correct, can it?' query was also pleading that 42 was the better answer to the how many tens in 42 question.

It's definitely not the end of the world - but as a parent I prefer the teacher who distinguishes place value from division (diennes).

diennes can be used to demonstrate all those things can't they? They arent a single function tool.

It's perfectly likely that the teacher was checking the answer against a sheet rather than staying that there was only one possible answer. The question is definitely ambiguous. As a secondary maths teacher I'd be looking for an answer of 42.7. A well written question should faced the same answer regardless of its audience.

But there are 42 tens in 427!

The concept of "ten" in terms of place value is just notation. It's just a convenient way to represent the numbers imho. In terms of the actual maths, there are ten tens in every hundred. I think OP's dd has demonstrated she understands the maths very well.

V poorly worded question imho. I speak as a Maths graduate btw.