Thought for the day: Be careful not to be too attached to your own observations and questions, especially at first. Students’ ideas may be very different!

Concepts: counting patterns; even-odd patterns; properties of numbers and operations; rules; doubling

(T): top list (M) middle list (B) bottom list (A) all lists
Note: In most cases, I would recommend showing these patterns one at a time, especially for younger students.

Beginning

I notice that the numbers are all even. (T)
I notice that it is like counting by 2s with some numbers left out. (T)
I notice that the “counting by 4” numbers are left out. (T)

I notice that the numbers go in a pattern of odd, even, odd, even, odd, even, … (M)
I notice that the numbers go in a pattern of up, down, up, down, up, down, … (M)
I notice that the number 2 is missing. (M)

I notice that all of the numbers are even except the first one (5). (B)
I wonder why this happens. (B)
I notice that the pattern goes up, down, up, down, …, but the jumps up are are bigger than in the middle list. (B)

I wonder how to predict what comes next. (A)
I wonder if the lists can go on forever. (A)

Exploring

I notice that the numbers go up by 4 every time. (T)
I notice that every “counting by 4” number is an even number. (T)
i notice a repeating pattern of 2, 6, 0, 4, 8 in the ones place. (T)
I notice that the tens digits go 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, etc. (two then three, two then three, etc.) (T)
I wonder why no numbers in the list are odd. (T)

I notice that the numbers go up 3, down 1, up 3 down 1, etc. (M)
I notice that if I skip every other number, it counts by the odd numbers: 1, 3, 5, 7, …. (M)
I notice that the rest of the numbers are the even numbers: 4, 6, 8, 10, …, but 2 is left out! (M)
I notice that I could put a 2 before the 1, and the patterns would still work. (M)
I wonder if every whole number will be in the list somewhere (if I keep it going). (M)
I wonder if any number will ever be in the list twice. (M)

I notice that the numbers go up a lot (different amounts), then down 4, over and over. (T)
I notice that 12 is in the list twice. (T)
I wonder if other numbers are in the list twice. (T)
I wonder if there will ever be any more odd numbers in the list. (T)
I notice that the jumps up are by the amount of the number itself (the number is added to itself or doubled). (T)

Creating

As they notice and wonder, students may create new patterns and multiple ways of describing the existing patterns . Many of their creations may flow out of things that they have wondered about. For example, they may

Create rules for continuing the patterns. (There will usually be more than one way to do this.)
Create new patterns by modifying the existing ones.
Create new patterns using their own rules.
Create pictures that illustrate the patterns.
Create stories to fit the patterns.

Reflecting and Extending

I notice that some patterns can on forever.
I notice that there can more than one way to describe a pattern.
I notice that it helps to think about how numbers change when I look at patterns.
I wonder if I can make up my own patterns that are even more complicated than these.
I wonder if I can extend any of these patterns backwards.

Notes

I thought of the first pattern as starting at 2 and adding 4 each time. (How do the lists compare if you start at 3 or some of number instead?) Some students may think of writing down even numbers and crossing out every other number starting a 4 (the multiples of 4).

I thought of the second pattern as starting a 1 and going “up 3, down 1, up 3, down 1, etc.). Later, I noticed that when I did this, every other number starting at 1 formed a list of the odd numbers 1, 3, 5, 7, … and every other number starting at 4 made a list of the even numbers, but 2 was left out. Then I realized that I could have put 2 at the beginning of the list and then gone down 1, up 3, down 1, up 3, etc. Some students may see that they can count in the list by jumping forwards and backwards and that no (whole) number except 2 will be left out.

I thought of the bottom list as starting at 5 and then following a pattern of “double, subtract 4, double, subtract 4, etc.). I noticed afterward that every other number starting at 5 looks like 5, 6, 8, 12, 20, etc. It looks like these numbers are going up by 1, 2, 4, 8, … an amount that doubles each time. This happens with the remaining numbers, too, except that they start going up by 2: 10, 12, 16, 24, 40, etc. I wonder why these things happen and if they will continue? I also wonder if 12 is the only number in the list that will ever repeat. I haven’t explored these questions yet, but I will soon, and I hope you do, too!

No matter how many ideas I list here, I am sure that you and your students will come up with tons of things that I never thought of. Let me know what you discover!