Thought for the day: Some prompts have specific problems hiding beneath the surface, but sometimes the questions and problems that students create may be as good or better than the ones suggested in these notes!

Beginning

Note: The image shown here is only part of the complete image, which has a number line running from 0 to 20. The notes below refer to the complete image.

I notice a number line with red Xs on it.
I notice that the number line goes from 0 to 20.
i notice that there are 8 Xs.
I wonder what the red Xs mean.
I wonder why it says “skip-counting” at the top.
I wonder if I could skip-count all the way across the line without hitting an X.

Exploring

I notice that I have to be careful not to count 0 as a jump when I start there.
I notice
that I cannot jump by 2s, because I would land on 6, and it has an X.
I notice that I cannot jump by 3s for the same reason.
I notice that I can go farther if I jump by 4s, but it still doesn’t work.
I wonder if I could get all the way across by starting at 1, 2, 3 or 4 instead of 0.

Note: After students have explored for a while, you might suggest that they try to get all the way across (from a number before 5 to a number after 17) in exactly 3 jumps.

Creating

As they notice and wonder, students may create and explore their own questions and puzzles like the one in the prompt. Many of their creations may flow out of things that they have wondered about. For example, they may

Create a story to go with this prompt. (mathpickle.com has a Tortoise and Hare story for it!)
Create new number lines with Xs in different places.
Create new rules about the number of jumps you take to get across.
Try to create lines that have only one solution for getting across.

Reflecting and Extending

I notice that there is more than one way to get across the line in 3 jumps.
I notice
that the small openings in the middle of the line make it harder to find answers.
i wonder if it is possible to land on every gap between the Xs.
I wonder what would happen if I made it a rule that you had to land on a certain number(s).
I notice that skip-counting by 2 can never work when some Xs are on even numbers and some are on odd numbers.
I wonder how many answers there are if I try to go across in 2 jumps? Or 4 jumps?
I wonder what is the smallest number of Xs I can put on the line so that there is no way to get across the line by skip-counting.

Notes

This prompt was inspired by the wonderful Tortoise and Hare problem from mathpickle.com. The notes above involve small variations on their problems, and the Math pickle activity eventually takes the idea much further and deeper. By the way, Math Pickle has an amazing collection of deep math problems for all ages. You should check them out! Like many of their problems, this investigation could become challenging even for professional mathematicians if you ask the right questions!

There are two ways to get across in 3 jumps:

Start at 1 and jump by 6s. (1, 7, 13, 19)
Start at 3 and jump by 5s. (3, 8, 13, 18)

(You would need to have a 22 on the line in order to start at 1 and jump by 7s.)

The real fun begins when students begin creating their own number lines and their own rules! Have them share puzzles and try to solve each other’s challenges!