Grammithmatic Part 2
If you want to read Part 1 where I discuss the number words we have for the TEENS and TENS, click here.
For Part 2 of tackling number naming conventions, another issue that stands in the way of many students truly understanding the base 10 system is the role of the zero, and what we call it.
Zero is a quantity, a benchmark, and placeholder. All students learn what zero is as a quantity. They connect 0 to the idea of having nothing very easily. They also catch on to the 0+n=n equation pretty fast; not just the idea of adding to ZERO but the formula itself, even at 8 years old. ZERO as nothing is easy.
As a benchmark, it’s iffy. It applies mostly to negative numbers which baffle teenagers sometimes. However, talks about rulers, scales, and centigrade thermometers can help students to at least see that we generally start counting from zero.
What is really tough is when it comes to place value, especially when borrowing and carrying in columnar adding, and regrouping when applying the distributive property. It’s overwhelming sometimes. And yet, without a ZERO, our number system is impossible. As such, it is imperative that students gain a strong understanding of how the ZERO works and what it means. Here’s a distracting link to Schoolhouse Rock you can show your class-but preview it first because you never know how people muck around with stuff.
So, how do we overcome some of this confusion?
Be aware the ZERO is there, and make sure students understand that it means something. When we have a number like 20, or 304, or even 2, the ZEROs go unmentioned, like they aren’t there; but they are, they have to be.
It’s not actually 20. It’s not even Two TENs. It’s Two TENs and ZERO ONEs.
It’s not 304. It’s Three HUNDREDs and ZERO TENs and Four ONEs.
The simple act of explicitly bringing the ZERO to a student’s attention, and giving a name to the position of the ZERO makes place value much more obvious.
Think of that ZERO as an empty box. Whenever we see a ZERO, there is a place we can put another number. We don’t think about it because we don’t say it (it takes too long to do such a thing practically, but for educational purposes…). If you think about it, even the number 2 has ZEROs. There are infinite ZEROs both before and after that 2, but for brevity and clarity’s sake we don’t mention them.
What’s important to understand is that empty boxes can be filled. You have experienced children who write 198, 199, 200, followed by 2001, 2002. This is because they have misunderstood that the ZEROs are placeholders for numbers that can be added to the HUNDREDs. They think that the ’00’ works like a plural S or some kind of suffix. It must be made clear that the first 0 after the HUNDRED is for any TENs that might come along, and the next ZERO is for any ONEs. What’s nice is the 0 looks like a cozy little container you can put stuff in.Again, labelling the ZEROs with their place value will support this understanding.
Here we have 200…Two HUNDREDs and ZERO TENs and ZERO ONEs. As well, we have Two ONEs. I’m going to add those ONEs in the empty ONEs container of 200 to make Two HUNDREDs and ZERO TENs and Two ONES.
I have Given ZERO a name. I know, ZERO is its name. So is nil, and cipher, and nought. But that’s not what I mean. If you think about 3, it can be THREE, it can be THIRteen, it can be THIRty, or THREE HUNDRED. And each name for 3 implies it’s location in the place value system (again, follow the link to Part 1 to see how to make this explicit). Well, I think we should do the same for ZERO.
So, we have the number 0 alone, and that is called ZERO.
What about for 10? First, we break 10 down and say it is One TEN and Zero ONEs. Then, when that is understood, or maybe in order to make sure it’s understood, we take things a step further, and name it One TEN None.
Did you see what I did there? To clarify, 21 is Two TENs ONE. Therefore, 10 is One TEN NONE. The TENS are no longer seen solely as large collections of ONES. By naming the empty ONEs column, we make the students aware a number is there, it has a purpose and a value even if its value is nothing.
If you keep this going, we have Twenty-None, Twenty-One, Twenty-Two….
We have Fifty-None, Fifty-One, Fifty-Two….
Next, 300 becomes Three HUNDRED, Empty-None. Get it? Fifty=Five TENS, Thirty=Three TENS, Empty=No TENS! (I’ve also toyed with Nonety or Nunty, but that might get confused too easily with Ninety. I’m still experimenting.) The point is, by naming the ZEROs, we emphasize the value of each part of the number as well as the ZERO’s placeholder role. This should prevent numbers like 60024 when writing about how many kids go to our school
Let’s do 2045? Two THOUSAND, NoHUNDRED, Forty-Five. Or perhaps you prefer None HUNDRED?
This can go on, but once we’re in the thousands, the irregularities of naming just repeat themselves. We, as a culture, haven’t used large numbers enough for the language to evolve shortened, lazier forms of big number names. How would you say 3 000 000?
Three Million, NoHundred Empty-None Thousand, NoHundred Empty-None.
Finally, I’ll give you a challenge that one of my students gave me. How would you write the number Empty-Six in numerals?
I’m pretty impressed that he understands that whole infinite ZERO thing, and I see this insight of his as proof that this idea of naming the ZEROs has some real legs. Let me know how it works for you if you try it, and tell me about any confusions or issues that popped up that I haven’t experienced yet.