Tuesday, May 9, 2017

1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, ...

Sequences are ordered progressions of numbers or other objects, as in the examples

1, 2, 4, 8, 16, …


A, B, AA, BB, AAA, BBB, …. 

In school mathematics, sequences are often called "patterns." I don't like that usage, because it's terribly limiting. Mathematical patterns can be found in visual designs, crystals, the multiplication table, or some totality of facts…patterns aren't just about sequences! You'll often hear people say, "Mathematics is the study of patterns." They don't mean, "Mathematics is the study of sequences."

My colleague William McCallum has a useful dictum: "Patterns are a tool, not a topic." For example, patterns in the multiplication table could be used as a tool for teaching about the properties of operations; patterns in the sequence 1, 2, 4, 8, 16 could be used as a tool for teaching about exponential functions. Whatever work is done with sequences at a given grade ought to transcend 'patternology' to intentionally build up students' strengths in the most important mathematical topics at each grade level.


Lately in idle moments I'll try to pass the time by thinking up a sequence of numbers that isn't yet included in the Online Encyclopedia of Integer Sequences (OEIS). It's a fun game—like trying to think of a notable topic that isn't in Wikipedia.

This was my first attempt:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 30, 41, 50, 61, 70, 81, 90, 111, 200, ...

The rule of this sequence is that, beginning with zero, each successive number must share no digits with the previous number—and be as small as possible subject to that constraint. For example, the number after 200 will be 311, because 311 is the smallest number greater than 200 that doesn't have 2 or 0 as a digit.

This is sequence A030283 in OEIS.

To create the sequence whose initial terms are in the title of this post, I considered a sequence of regular polygons with sides of unit length (triangle, square, pentagon, hexagon, etc.). Choosing a standard position for the polygons (centered at the origin), and also choosing a standard orientation (with the top edge horizontal), I counted how many grid points (m, n) were inside the polygon or on its boundary. (Here m and n are integers.)

Here are some diagrams showing the first eleven values 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13:

My sequence doesn't appear to be in OEIS. There's a partial match with A219844, but the two sequences aren't the same.


When you look at the numbers 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, what do you notice? Will the feature you noticed hold true for the entire sequence, out to infinity?

Feel free to write any observations, conjectures or arguments down in the comments. Also, feel free to put a sequence of your own in the comments!


My daughter and I talked a bit about sequences during Saturday School a few weeks ago. That morning, my daughter was writing in a workbook she'd brought home from school. Looking across the table, I noticed that whoever had written the workbook had misunderstood 4.OA.5, a standard that involves sequences. In an effort to rescue the math, I quickly sketched the following problem:

Start with 256. At each step, divide by 2. Repeat forever!

a) Show 4 steps of the pattern (sequence)

b) Prove or disprove: every number in this pattern (sequence) is a whole number.

(As you can see, I took this opportunity to reinforce fractions and calculation of quotients. Also, note that the rule for the sequence is given. This is a math problem, not an  IQ test.)

By making the starting number a large power of 2, I had hoped to lure her into believing that all of the numbers in the sequence would be whole numbers. That gambit didn't work, however—she saw from the outset that the values would eventually drop below 1, and she proceeded to show this by generating sufficiently many values. (Actually, more than enough values! I then drew her attention to the second of the two arguments below.)

My daughter had fun solving this, and we enjoyed talking about it. What's cooler than infinity? Done right, I think 4.OA.5 can prompt some very good mathematics. I expect it has also helped publishers and school systems hang onto old-style, non-aligned "pattern" work in the curriculum.


Ashley French said...

Hey Jason,

I apologize for not knowing where to post my comment. I have a question about "MACC.4.G.A.2- Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles." At face value, the standard appears that when working with this standard and identifying triangles that the focus should be identifying right triangles. However, the Progression Documents mention fourth graders "can identify these properties in two-dimensional figures. They can use side length to classify triangles as equilateral, equiangular, isosceles, or scalene; and can use angle size to classify them as acute, right, or obtuse. They then learn to cross-classify, for example, naming a shape as a right isosceles triangle." I know that the progression documents proceeded the standards, so I am not sure that this statement is still applicable. Could you please offer guidance as to whether 4th grade students should be identifying equilateral and isosceles triangles when studying this standard and when being assessed on this standard? Thank you so much for your support in helping our district "Achieve the Core".

Jason Zimba said...

Hi Ashley, thanks for your comment. One place where people have been able to ask detail-oriented questions about particular content standards is in the various forums on the commoncoretools blog. Here's the forum for K-6 geometry: http://commoncoretools.me/forums/forum/public/k-6-geometry/

I didn't see your question addressed there, but if it isn't there then you could always initiate a new thread.

I'll offer a reply here. Looking at one of the examples you quoted - namely, using side lengths to classify a triangle as equilateral, etc. - I think it is clear that the language of MACC.4.G.A.2 doesn't require this. So an assessment task that aligns specifically to this standard wouldn't hinge on identifying equilateral triangles & so on.

Some addenda to this answer:

1) To say that it isn't required isn't to say that it *mustn't* be done in grade 4, but that the implication of the wording of MACC.4.G.A.2 would be that it oughtn't be done in such a way, or to such an extent, that it compromises what *is* explicitly required (not only in the G domain but in grade 4 as a whole).

2) If you look carefully at the first paragraph in the Grade 4 section of the Progression document that you quoted from, I think you will see that the verbs change after the first sentence. In the first sentence, which tracks 4.G.A.2 and 4.G.A.3, the verbs are simple: students describe things, analyze things, compare things, etc. But in the next sentence (the one you quoted from), the verb changes: now "students *can* use side lengths ...." (emphasis added). By making this shift in language, the drafters of the Progression document appear to be gesturing at possibilities that curriculum designers might productively consider. I think this reading is consistent with my remarks in (1).