Math Is Elemental · Chapter 2

Rounding

Rounding is a type of estimating. Sometimes we don’t need all the details. With numbers, the details are the smaller place-value positions. We can ignore them, but we need to figure out how best to represent the part we are about to ignore.

This two-column setup follows the logic of place value and carrying. Slide it along the place-value chart, and the setup itself does the rounding. The target place-value column becomes the new counting digit. These are the new ‘1s’, as indicated by the large red 1.

The digit to its right shows how much of a new ‘1’ has been made. Match the green block count to this column's digit.

The question: “Did we almost make another 1?”

Example: Round 248,356 to the nearest ten thousand

What's the address?

First step: find the place value column.

The nice thing about the hundreds, tens, ones columns repeating themselves in three-column groups is you get to use the same words to describe numbers again and again - all the way to infinity. The catch: you have to listen for and use group names.

A three-column group name was mentioned: Thousand. That means you must leave the no-need-to-name Ones group and go to the must-name Thousands group. Once you’re at the correct group address, the Tens column is always in the middle, between the Ones and Hundreds. It's the same order every time - just get to the right group first.

The confusing part is that the group name comes last when we say it, but you need to go there first. It's like a street address: 248 Broadway. We say 248 first and Broadway second. But to find the house, you first find Broadway, then find 248.

Not driving yet? It's like city, state when you address a letter. Or adjective, noun when you write. Or room number, building name.

Sometimes the most important part comes last.

Rounding to the new counting unit

Now that we've found the place value column, make it our counting unit. Move the big 1 to the place value you are rounding to. That's what we're counting now. These are our new '1s'.

We normally count by actual 1s. Now we'll be counting by ten thousands. The whole place value chart to the right is just about to be reset to 0s.

Ten thousands are the new '1s' and the count is currently 24.

Match the number of small 1s to the digit immediately to its right.

Time to make a call...

Before we stop keeping track of these smaller places and replace them with zeros, how do we best represent their value? Is the leftover worth a carry or not?

Is 3 closer to 0 or to making another complete ‘1’? How about 8?

If so, count it. Fill the small blocks up to 10, then carry the 1 just like always. This is called "rounding up."

If not, clear it away. Ignore its count. It becomes 0. A placeholder. The count stays the same because there is nothing to carry. Called "rounding down."

5 is a tie
tie goes to the runner
(e for effort)

Number Line Plot unitizing sets up the number line

What is the roundest digit? (zeros)

Rounding is making zeros (‘rounds’) out of all the digits to the right of the column we are rounding to. The 4 in this case, so separate the number:

24 is the Count.
Ten thousand is the Name.
We are at 24 ten-thousands.

24 is the first number on the number line that we need to mark:

What comes after 24?

Remind me again, what are we counting? (ten thousands)
So the 24 needs to go in which columns?
And how many 0s do we need to place them there?

After the place value reasoning, students may notice a shortcut: when rounding, the number of zeros needed equals the number of digits we are setting to 0. We are setting four digits to 0, so we need four 0s to put 24 back in the correct place value columns.