Beginning

I notice that the picture shows every way to make three-digit numbers from 1, 4, and 7.
I notice that all of the numbers will fit onto the number line.
I notice that it is hard to put the numbers in exactly the right place.
I notice that the picture helps me to see which “counting by 100s number” each number is closest to.

Exploring

I wonder what would happen if I used three other digits.
I wonder if there is an easy way to see which “ counting by 100s number” a number is closest to without drawing a number line.

(Note: I will use the word “round” from here on out, but there is no need to introduce this vocabulary to students right away. Wait until they get comfortable idea first.)

I wonder if a picture like this could help me round to the nearest 10 or 1000.
I notice that there are always 6 ways to make 3-digit numbers from three different digits.
I notice that the tens digit tells me how the number is going to round to the nearest hundred.
I wonder what would happen if I make 4-digit numbers by putting a 0 in each ones place (1470, 1740, 4170, etc.)

Creating

As they notice and wonder, students may create new sets of numbers to test and different number lines for different situations. The things that they create may come from what they have noticed and wondered. For example, they may:

Create a variety of number lines to help with rounding.

Note: Grid paper comes in handy for making number lines. 100-boards (10 by 10 grids showing the whole numbers 1 - 100) can help, too!

Create lists of other numbers by rearranging the digits (possibly including 4-digit numbers).
Create tables to organize the numbers.
Create and observe patterns from the lists and tables.
Create rules for rounding numbers quickly (without a number line).

Reflecting and Extending

I notice that number lines can help me picture how large numbers are and what numbers they are close to.
I wonder if I could use a 100-board to see this, too.
I notice that patterns can help me discover shortcuts.
I notice that the tens digit helps me figure out how to round to the nearest hundred.
I notice that the hundreds digit helps me figure out how to round to the nearest thousand.
I wonder if I can always round to a place value by looking at the next smaller place value.
I wonder if I can round to other numbers like the nearest 5 or the nearest 6.
I wonder how I could change my number lines (or grids) to help me do that.

Notes

This prompt is built around an idea from the NRICH Maths website. They use dice to implement the task. For more detail, see nrich.maths.org/10426/note. NRICH Maths has an excellent selection of deep, challenging problems for students of all ages!

As noted in the “Thought for the day”, this prompt is great for guiding students to discover their own rules for rounding. They just need to know what rounding means in order to get started exploring. For example, rounding a number to the nearest hundred means finding the closest “counting by 100 number” to that number.

147 and 174 are between 100 and 200. 147 is closer to 100. 174 is closer to 200.
417 and 471 are between 400 and 500. 417 is closer to 400. 471 is closer to 500.
714 and 741 are between 700 and 800. 714 and 741 are both closer to 700.

By going through this process with many groups of 3-digit numbers, keeping track of their data, watching for patterns, and trying to understand what causes them, students may discover the shortcut of using the tens digit to decide whether to round up or down. They will not know what to do when the number is exactly in the middle. In this case, you can tell them that it is traditional to round up. (This is true in grade school and many other situations, but there are cases in which people use different rounding rules in this special situation!)

Once students understand the process with 3-digit numbers, they extend the problem by either putting 0s in the ones place:

1470 1740 4170 4710 7140 7410

or by choosing four 1-digit numbers and making a lot of 4-digit numbers out of them. (There will be 24 possibilities if all four digits are different, so students may not find all of them, but that’s okay!)

When they deal with larger numbers, students will need to think hard about what place value to round to and how to draw number lines in a way that helps them visualize the closer number.

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