Building Numbers
Where did the '0' come from?
It is always there! Believe it or not, at first it is useful to think of the 0s as still being there beside the original counting digits they place into position.
Teaching zero permanence. A mental model for building numbers. We want students to see that 500 is represented in 538 even when they can't see it displayed. This exercise introduces the idea that numbers are built from components.
The zero is irreplaceable because it can be replaced. Think of the 0s as belonging to the 500. The two 0s place the 5 in the hundreds column. When there is a counting digit to show, it takes the 0's place.
1-9 do the counting. Zeros do the placing.
Zeros are like the spaces between these words. The analogy isn’t a stretch, since zero started out as a blank space and evolved long after the digits 1 through 9. The difference is that in math, the spaces are also placeholders.
(Zeros don't really hide under other digits, as far as we know, but we can address that later.)
If you can count to 9, you can build to 999.
Write 1 - 9 in a column and repeat the numbers together. Then, put a 0 after each digit using a different color. This new set of numbers 'rhymes'. Repeat together until the student becomes the teacher. Next, add another 0, keeping the 0s the same color, and..more mimicry. Building the columns like this helps to show how the counting pattern stays the same.
For younger students, start with ones and tens only. Skip the teens initially because the names are irregular and blur the pattern.
When you eventually introduce the teens, explain that "teen" is an old word for "ten." Once they think of teen as ten, they will more easily see this inverted naming pattern, where the ones come before the tens (e.g., four-teen meaning four and ten). Exceptions to these exceptions: eleven and twelve. These come from old words meaning "one left over" and "two left over" after counting to ten.
Time to pick a number and build it. Use 3D toy digits if possible. Print columns: 100 to 900, 10 to 90, and 1 to 9. Leave a margin between the columns, or cut and separate them, but keep the columns in this order as a reference for building numbers: Hundreds | Tens | Ones, or use the attached template.
Say the first part of the number, then wait until you assemble it before saying the next part. Build each digit with its respective 0s (one color for each digit and its 0s). For 538, say,
"Five hundred" (build 500)...
"thirty" (build the 30 over the two 0s that place the 5)...
"eight" (8 is placed over what is now two 0s)
Now, disassemble the 538 to show the 500, 30, and 8 separately.
Repeat the cycle with 538, then build some other numbers. Writing and reading numbers can begin before students fully understand every place-value relationship. As with learning to talk, students imitate the form first and understand it more deeply later.
Next, assemble a 'flat' 538. That is, place no zeros under the digits.
…but the zeros are still there, right?
Point to the 5 and discuss the name of this Place Value position, how it can be represented or modeled.
Surprisingly, it's best to start with the hundreds because the spoken words already give the unit: five hundred = five 100s.
After the connection is made, ask the student to draw it: five circles labeled 100. After practicing with a few hundreds, move to the tens. If the student needs a hint for the 3 in 30, move it to the hundreds place: “What would the 3 mean there?” Once the student answers 300, return it to the tens place and ask them to name 30 correctly.
If necessary, practice the tens column rehearsed together earlier. After the student names the position and can represent it as three circles labeled 10, complete 538 with eight circles labeled 1.
This takes repetition, but so does learning to speak. The lesson introduces place value and the idea that numbers are built from components without naming either concept. Like learning to speak before learning grammar. All you need to begin is the ability to count to 9.
0notes
Zero is not just a placeholder. It is the point everything else is measured from on the number line. The origin.
Also, multiplying by zero: 0 × any number is 0.
“I want 0 copies of 5, please.” Okay, no copies, coming right up. Works when you switch the order, too. “Scratch that, can you make me 5 copies of nothing?”
You can see why no one ever does this calculation. “Hon, you know how you bought me 0 carrots at the store last week? I want 5 times as many this week.” Imagine the grocery list.
Lastly, what is a number divided by zero? No one ever asks this either, but if they do, say it’s undefined. It may feel like the answer should be infinity, but zero is not a tiny piece. Zero means no piece at all. No number on the planet divided by zero produces an answer. There is..no answer. Undefined.