Archive for the ‘Prime Factorization’ Category

More about Mathematical Knitting

Saturday, August 24th, 2013

On Wednesday, I finished knitting a Prime Factorization Blanket for my new little niece.

In my post about the blanket, I explained how the colors show the prime factorization of each number from 2 to 99. But I didn’t talk about the patterns, and I want to say a little bit about that here.

In fact, the only reason the Prime Factorization Blanket isn’t quite as good as the Prime Factorization Sweater is that I can’t have rows of 8 on the back and rows of 2 and 3 on the sleeves.

And the Prime Factorization Scarf is good for getting the flow of the numbers.

However, I do think the patterns in the 10 by 10 grid are a little easier to see with the larger diamonds on the blanket. Here’s the complete blanket laid out:

Let’s start by looking at the diagonals. 11 is 1 bigger than the base of 10. So the color for 11, red, goes in a diagonal across the blanket from the bottom left to the top right.

9 is 1 less than the base of 10. 9 = 3 x 3, so every number with a factor of 9 has two sections of yellow, the color for 3. You can see the yellows going diagonally up the blanket from the bottom right to the top left.

Oh, and I nearly forgot the more obvious ones. Since 2 and 5 are multiples of 10, they line up in columns. Every second column has turquoise for 2, and every fifth has green for 5.

Once you’re used to focusing on one color, you can pick any color and watch how it distributes evenly around the blanket. Take 19 for example, dark pink. You can see it climb up the blanket from the bottom right to the top left on a steeper diagonal than the one for 9.

And if you look at the colors it’s paired with, first it matches with 2, then with 3, then with two sets of 2, then with 5.

Another fun pattern is that the columns are sets of numbers that are congruent mod 10. So if you add, subtract, multiply or divide any two numbers in the same columns, your result will be in the same column.

For example, 2 + 17 = 19. Well, 62 + 27 = 99. The numbers in the second equation are from the same columns as the numbers in the first equation.

And that’s only the beginning of the patterns you can find.

Now that I’ve mailed off the blanket, I’m consoling myself by getting excited about the Pascal’s Triangle Shawl I’m going to make.

Pascal’s Triangle is formed by starting with 1, then adding a row of numbers where each number below is the sum of the two numbers above it.

Here’s how it works:

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
1 11 55 165 330 462 462 330 165 55 11 1
1 12 66 220 495 792 924 792 495 220 66 12 1
1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1
1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

The cool thing? Even though you’ve got really big numbers, when you factor them, the only prime factors are ones that have already appeared in the triangle. So I only need colors for the prime factors 2, 3, 5, 7, 11, and 13.

I’m stopping at row 15, because on row 16, we’ve got numbers with more than 6 prime factors, and that’s the most I’m prepared to accommodate unless I make my diamonds bigger. But up to 15 is going to be lovely.

I chose colors yesterday and tried to order the same colors. I’m not sure the ones I have aren’t discontinued colors by now, but when my order arrives, I can get going. (In fact, I’ve already started with the colors I have, hoping I won’t have to take out too much.) Here’s the color scheme I chose, using Cotton Classic yarn by Tahki.

2 is on the bottom left, and then it goes around counterclockwise. So 2 will be pale pink, 3 will be rose, 5 will be red, 7 will be purple, 11 will be yellow, and 13 will be turquoise. And they are going to repeat in beautiful ways, just you wait!

I’ve already begun, though if it turns out that the pink I’ve ordered is a different shade from what I have, I’ll have to take out the square for 2 that I’ve begun. But I can’t stand waiting for the order!

You can see there the initial diamond for the first row: 1.

Second row has two diamonds for the second row: 1 1

Third row, I’ve knitted the first diamond for 1, and I’ve begun the next diamond, for 2.

When I get to the row with 4, I will start showing the prime factorization, so 4 will be listed as 2 x 2, with two sections of pink.

The way I’ll show the prime factorization will be exactly like the blanket, but the patterns will be very different, always with the diamond representing the sum of the two numbers on its lower edges.

And it will get cool on the top edge with numbers like 6435 = 3 x 3 x 5 x 11 x 13

I can’t wait to show pictures of the final result. I think it will be beautiful!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

The Prime Factorization Blanket!

Wednesday, August 21st, 2013

Yes! My Masterpiece is finished!

What is this, you ask? This is a Prime Factorization Blanket!

With colors, it shows the prime factorization of all the integers from 1 to 99.

Here is the entire blanket, laid out flat:

Here’s how it works: Every prime number gets a color. The numbers start in the lower left corner.
I left a space for 0.
1 is the background color, white.
Then the next color is 2, a prime, so it gets its own color, blue.
3 is prime, and gets its own color, yellow.
4 is 2 x 2, so that square is two sections of blue. (You can tell on the blanket that there are two sections.)
5 is prime, and gets a new color, green.
6 = 2 x 3, so that square is part blue and part yellow. And so on.

I’ve got 0 through 9 on the first row, 10 through 19 in the next row, then 20 through 29, and so on through the top row, which is 90 through 99.

To show it more clearly, let’s look at each quadrant. Here’s the bottom left quadrant:

I put in the factors for each color. (After a few colors, I stopped putting in the “x” symbol for times.) I put a reference number on the left side so you can easily see which row. This set has 1 through 4, 10 through 14, 20 through 24, 30 through 34, and 40 through 44.

Now let’s look at the bottom right quadrant:

This picture shows 5 through 9, 15 through 19, 25 through 29, 35 through 39, and 45 through 49. For example, see if you can spot 48, which has a prime factorization of 2 x 2 x 2 x 2 x 3. Or look at 38, right below it, which equals 2 x 19.

By the way, this blanket is for my little niece, the daughter of my brother, who is, if it’s possible, even more of a math geek than me. On the 17th of December, my sister-in-law had an ultrasound, and we learned that the baby would be a girl, so I chose shades of pink for the next primes that came up, 17 and 19!

Now here’s the upper left quadrant:

This picture shows 50-54, 60-64, 70-74, 80-84, and 90-94. Can you find 62 = 2 x 31? Or 94 = 2 x 47? (I have to note that the colors are more distinct in person, and you can tell by the garter ridges how many sections there are of each color.)

And finally, the upper right quadrant:

And this, of course, covers 55-59, 65-69, 75-79, 85-89, and 95-99.

I’m so happy to finish it! The yarn is the same as what I used for my Prime Factorization Sweater, Cotton Classic. This yarn has enough colors (most important qualification), and it’s wonderfully soft — perfect for a baby blanket. I used a lot of leftover colors from the sweater, in fact.

The only really hard part? Giving it away! But I got the *idea* because my brother’s wife was having a baby, so this seems only fair to send it to the baby, as promised. Unfortunately, she lives on the other side of the country — so the one stipulation is they must take *lots* of pictures of her with it!

In fact, I thought of a way to console myself for giving away the blanket. My next project will be a Pascal’s Triangle Shawl!

I tested out, and the shape will work great!

I loved doing the entrelac squares for the blanket — it was much much easier than the intarsia I used on the Prime Factorization Sweater. And it will be easy-peasy to make a triangle instead of a square. I’ll use factors and do Pascal’s Triangle…. More on this to come, you can be sure!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Oh, and don’t forget! If you want your own prime factorization t-shirt or tote bag, you can find them at my Cafepress shop.

Halfway on the Prime Factorization Blanket!

Friday, April 12th, 2013

I’m halfway finished with my new niece’s Prime Factorization Blanket!

That’s the good news. The bad news is that it doesn’t look like I’m going to finish before Arianna arrives. But the good thing about a blanket is that it doesn’t have a size, right?

I actually finished up to 49 a week ago. I did not take a picture and report my progress. And then — when I started on the next row and began knitting 50 — I discovered I had used the wrong color when I knitted 40! I had used the color for 3 in place of the color for 5! *shudder*

(Those who need to be brought up to speed, I explain the blanket in previous Prime Factorization posts.) 40 should be 2 x 2 x 2 x 5, so the square was divided into four sections, with three of them the color for 2 (blue), and one the color for 5 (green). But, horror of horrors, I had used the color for 3 (yellow)! And I didn’t even discover it until I after I had knitted 50 = 2 x 5 x 5, also using yellow when I should have been using green! Yikes! But then I was getting ready to start on 51 = 3 x 17, and then it dawned on me that the color for 3 is yellow, so it is NOT the color for 5, and I’d been doing it wrong!

Fortunately, it was on the end of the row, and relatively easy to fix. I took out the knitting I’d done on 50, undid the last white square in the row, and then undid just the yellow section I’d put on the top of 40. I replaced it with green, knitted the white square back, and then started up on 50.

But I’ve decided that taking a picture after each row and *checking* the numbers carefully is a good idea!

I did discover a mistake when I finished knitting my prime factorization sweater. I think it was in the rectangle for 48. But 48 has five factors, and in that piece of knitting the factor for three only was about four stitches. So I was able to pick them out and put in the correct color with a crochet hook! It would not be so easy to do on this blanket, so I am going to have to be more vigilant!

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Librarians Help – With Math!

Tuesday, March 26th, 2013

Today I had my Colors and Codes program that I mentioned last week.

Now, I spent ten years of my life teaching college math, but doing math programs at the library is so much more fun!

Why? Well, the biggie is I don’t have to grade them, so it makes the whole thing much more light-hearted. I’m showing them things about math that I think are really cool, and they get to think about ways to do it themselves. And it’s all just for fun. At the library, we teach people things they want to know! If they don’t want to know them, they don’t need to come. It’s that simple!

Here’s what I did. I showed the kids my prime factorization sweater (wore it of course), and we worked out how it works. (That was fun!) I told them if colors can represent numbers, they can also represent letters. Just use 1 to 26 for A to Z. So you can write messages this way. I showed them a prime factorization code, then showed them other bases and how you can make codes with them. We wrapped it up by getting out sticky foam shapes and they could put a coded message or just a pretty pattern around a picture frame or on a bookmark or a door hanger.

The highlight for me, I think, was when a girl was working on coloring in the prime factorization chart on the hand-out. She was stuck on 24. I asked her what it equaled, and she said 12 x 2. So we looked at 12, and then the light went on and we talked about how you could do figure it out different ways, but you always got three 2s and one 3.

Now, I’m going to write some notes to myself while the program’s fresh in my mind. It went well; the kids had fun. But I want to do it again this summer, and hope it will go even better.

1. I’ll set the age level higher. I do think I lost a few kids this time. I think I’ll set it at 10 or 11 years old rather than 8. You want the kids to be fully comfortable with multiplying. Now that I think about it, when I did this program a few years ago at Herndon Fortnightly Library, I think the age limit was 10.

2. We’ll do some coloring on the prime factorization chart before I move on. This group did work out with me how it works. I didn’t want to get bogged down, but I think some coloring would help them understand it better.

3. I’ll have them figure out the numbers for their name in every code I go over. For example, my name, Sondy, in a base 10 code is 1915140425. (S is the 19th letter, O is the 15th, and so on.) In a prime factorization code, it’s 19 1 3 5 1 2 7 1 2 2 1 5 5 1. (19 x 1, 3 x 5 x 1, 2 x 7 x 1, 2 x 2 x 1, 5 x 5 x 1) In a Base 6 code, it’s 3123220441. In a Base 5 code, it’s 34 30 24 4 100. In Binary, it’s 10011 1111 1110 100 11001. Taking the time to do that would mean they’d get what I was having them do when they went to use the foam sticky pieces.

4. We’d do some coloring on the other charts before we moved to the foam shapes. Then I’d have them do their name with the colors they picked.

5. I’d show them exactly how I did my name on the bookmarks, one using colors and one using shapes.

Did I mention everyone did have a good time? But I think I’ll do a little more hands-on, using their names, before I move to the craft next time.

But it was a great trial run!

And don’t forget! Librarians help! We get to show kids how much FUN Math is! And we don’t even have to test them on it!

Colors and Codes

Tuesday, March 19th, 2013

I just got a tweet that made me prouder than I’ve EVER been of my Prime Factorization Sweater, and that’s saying a lot.

The tweet was from @milesmac, Miles MacFarlane, a teacher, with the words, “#LeilaN students deciphered @Sonderbooks Prime Factorization Sweater – Now making own code #7Oaks”

Here’s the picture that accompanied it. Even by the backs of their heads, you can tell those are engaged kids!

Yes! That’s what it’s about! Mr. MacFarlane, you made my day!

And the timing is lovely. Next week, at my own City of Fairfax Regional Library, I’m doing a program I’m calling “Colors and Codes” where we’re going to do exactly that. I’ll wear the sweater (or maybe my prime factorization t-shirt and bring the sweater. And the scarf). I’ll show them how we can assign each letter of the alphabet a number from 1 to 26. We’ll start with a factorization code, but move on to things like Base 6 or Binary. And I’m going to have foam shapes for them to make crafts with codes in colors or shapes.

Yay! See, we don’t have to make Math fun! Math is fun!

Prime Factorization Progress – To 39

Tuesday, February 26th, 2013

I’ve already posted several times about my Prime Factorization Knitting, and I can’t resist posting pictures every time I get another row of numbers done on my new niece’s Prime Factorization Blanket.

You can get more detail of how it works in the earlier posts, but basically each prime number gets a color, and each number gets a square divided into the colors for the factors of that number. I’ve finished up to 39. (I’m not putting an exclamation mark after that statement, since I haven’t gotten to 39 factorial.) Here’s how the blanket looks so far:

And here’s a close-up on each side, with the numbers written in. You’ll have to figure out the factors. And I can assure you that it’s a lot easier to tell when there are two or three (or four or five) of the same factors in one number when you can see and feel the blanket. I divided it with garter ridges, and the photo couldn’t really catch that.

Here’s the left half:

And the right half:

Don’t forget that you can get your very own Prime Factorization T-shirt at my Cafe Press shop for a lot less effort than this blanket is taking (but okay, you won’t have as much fun as I’m having). I took it to a Youth Services Librarian meeting today, and only the unwary asked what it was going to be. I must admit, it’s a lot better for knitting during meetings when I’m on one of the white rows.

Will I finish before Baby’s arrival in May? I hope I will at least be close….

Prime Factorization Blanket – to 29

Monday, January 21st, 2013

I got done another row of numbers on the Prime Factorization Blanket for my arriving niece!

It’s hard to see the ridges in the solid colors, so here are close-ups of the left half, then right half:

The bottom row in the picture is 1 (white), 2 (blue), 3 (yellow), 4 = 2 x 2, 5 (green).

The second row is 10 = 2 x 5, 11 (red), 12 = 2 x 2 x 3, 13 (tan), 14 = 2 x 7, 15 = 3 x 5.

The top row is 20 = 2 x 2 x 5, 21 = 3 x 7, 22 = 2 x 11, 23 (baby blue), 24 = 2 x 2 x 2 x 3, 25 = 5 x 5.

Now the right half:

Here we have the bottom row of 5 (green), 6 = 2 x 3, 7 (dark purple), 8 = 2 x 2 x 2, 9 = 3 x 3

The second row is 15 = 3 x 5, 16 = 2 x 2 x 2 x 2, 17 (pink), 18 = 2 x 3 x 3, 19 (dark pink).

The third row is 25 = 5 x 5, 26 = 2 x 13, 27 = 3 x 3 x 3, 28 = 2 x 2 x 7, 29 (periwinkle)

I really like the way it’s turning out!

You can read more about my prime factorization knitting in previous blog posts or via my Pinterest board. And don’t forget to look in my cafe press shop for prime factorization t-shirts.

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Prime Factorization Blanket – Second Row

Friday, December 21st, 2012

I’ve finished the second row of numbers (third row of rectangles) in my Prime Factorization Blanket!

The fun part was that my brother and his wife found out on the 17th of December that their baby is a girl. So, since I was coming up on the prime number 17, I chose to use pastel pink to represent 17. For good measure, I used a pretty rose pink to represent 19.

I only hope that having all that turquoise blue won’t make people think it’s a blanket for a boy, but I’m hoping it’s multicolored enough, it won’t give that idea.

I couldn’t manage to write in all the numbers on the picture, like I did after the first row, but in real life I assure you, you can tell when there are two factors of the same prime.

So here’s how you read the blanket:

The bottom row starts with a blank space for 0.

1 is the same as the background color, since 1 is a factor of every number.

2 is turquoise blue.

3 is yellow.

4 = 2 x 2, so two sections of turquoise.

5 is green.

6 = 2 x 3, so a section of turquoise and a section of yellow.

7 is purple.

8 = 2 x 2 x 2, so three sections of turquoise.

9 = 3 x 3, so two sections of yellow.

Then I did a row of white rectangles (diamonds). Second row of color:

11 is red.

12 = 2 x 2 x 3, so two sections of turquoise and one of yellow.

13 is brown.

14 = 2 x 7, so a section of turquoise and a section of purple.

15 = 3 x 5, so a section of yellow and a section of green.

16 = 2 x 2 x 2 x 2, so four sections of turquoise.

17 is Pink!

18 = 2 x 3 x 3, so a section of turquoise and two sections of yellow.

19 is rose pink.

Next I’ll do a row of white rectangles, then start the next row with 20. The primes in that row will be 23 and 29, so I’ll have to bring in two new colors.

The color sections will show up better after I’ve knitted the white rectangles, but I was impatient to show what I’ve done!

I’m very pleased with how this is turning out. I may have to make myself a Pascal’s Triangle Shawl when I’m done….

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.

Bedtime Math!

Wednesday, December 19th, 2012

I’m so excited! Today, thanks to a note in the ALSC (Association for Library Service to Children) newsletter, I found out about Bedtime Math.

Why do I think Bedtime Math is so awesome? Because that’s totally what I did with my younger son.

My first Master’s degree was in Math, and I was a college math instructor for ten years. College students in general ed math classes are generally not excited about math. So when we started doing math problems with my excited son at bedtime — I’m not sure how it started — my son quickly learned those magic words I absolutely COULD NOT resist — “Just one more math problem, Mommy, please!” He could extend bedtime forever with those magic words.

I don’t remember how it got started, but I do know that we were in the thick of this when he was 5 years old. His brother turned 12 years old in March. I turned 36 in June. Sometime in there, I told him that when he turned 6, then his age plus his age would equal his brother’s age. But, even better, his age TIMES his age would equal my age. His next question was pretty natural, “What’s TIMES?”

One week later, his brother asked him a problem I never would have tried: “What’s 16 times 4?” Timmy (the 5-year-old) figured it out *in his head.* Without knowing times tables. So that was the context of “One more math problem, Mommy, please.” I’d give him progressively harder addition problems — and then it got to be progressively harder multiplication problems. All done in his head, at bedtime. For fun.

Of course, it all starts with counting. I remember with my older son, just counting as high as he could go in the car while running errands. It’s fun when they really realize how it works and that they could go on and on forever. He was also the one who kept making up words for “numbers bigger than infinity.” I couldn’t quite convince him that didn’t work.

(Now my younger son, a Freshman at the College of William & Mary, recently spent his free time devising an algorithm to choose a completely random book from all the volumes in the campus library. That’s my boy!)

In my current job as a librarian, I was thinking about all the counting and math we did when my kids were small. And then thinking about the Every Child Ready to Read workshops, where we encourage parents to read, talk, write, sing, and play with their kids. I’m going to do the workshop “Fun With Science and Math for Parents and Children” — only I think I’m going to take out the Science and just focus on Math.

See, the thing is, I don’t believe for a second you have to “make” Math fun. I think math *IS* fun, and children naturally think so, too. Can I communicate that to parents?

I’m also planning to do a program with older kids about using math to make coded messages with colors or shapes. It uses ideas from my Prime Factorization Sweater and my Coded Blessing Blanket. I did the program a few years ago, a little afraid I’d lose the kids, and they totally loved it.

All this is to say: Bedtime Math! YES! I can present this as an idea for parents who need help thinking of problems to talk about with their kids, who might not think them up as easily as I did. (I also taught my kids the chain rule in calculus because I wanted to teach it to someone who would get it right. But I don’t think I’ll recommend that to parents.)

I still say, as a librarian, part of my job is the FUN side of learning. At libraries, we help people find information to teach themselves. But in the children’s department, a huge part of our job is helping parents make learning a natural and fun part of their family life. We don’t have to test them! We don’t have to follow the book or the curriculum! We can show them ways to think about the concepts that are just plain fun!

I’m going to be looking for more articles about early learning of mathematics. I think it can fit in nicely with Early Literacy Skills that we emphasize so much. But mostly I’m jazzed. Other Moms are going to hear those magic words: “Just one more math problem, Mommy, please!”

Prime Factorization Blanket – First Row

Friday, December 7th, 2012

So, Tuesday I posted my plan for making a Prime Factorization Blanket for my new niece or nephew. I then learned that my brother and his wife may know the baby’s gender by December 17th. Did I want to wait that long to choose the colors and start?

Short answer: No. I got to thinking: It’s not like this won’t be a very multi-colored blanket. I had thought about using shades of blue or shades of pink at the beginning, but I don’t think that’s a good plan. Since the colors represent numbers, and since a baby’s going to see this, better to have distinct colors with distinct names as the prime factors that show up most often.

On top of that, I happen to have a full skein and more of a turquoise blue left over from another project. Turquoise worked out very well as the color for 2 in my prime factorization scarf. It doesn’t cry out “boy,” but neither is it a bad color for a girl. And best of all, it goes well with pretty much every other color. (And 2 has to do that.) I decided to go with bright, rich colors for the primes that will be most predominant.

Thanks to a fairly long management meeting and a day off today where I needed to read, I’ve already finished the first row. I’m very happy with the colors! Now, the first row consists of just knitting 9 squares. The next row of white will convert them into diamonds. I’m also proud to say that I sewed in the starting ends of all the yarn. And that’s my plan to go on with: At the end of each row, I’ll sew in all the ends that were loose at the start of that row. (I don’t want to sew in the ends right next to a live stitch.) That way, it won’t be a daunting task at the end of the blanket.

So here’s the first row, the numbers 1 through 9 (0 is a blank space.), with the numbers the colors represent written on the picture:

I admit it’s getting where I’m going to have a hard time giving this away! Good thing I’ve publicly said it’s for my new niece or nephew!

And that does remind me. If you’d like your own Prime Factorization T-shirt or Tote Bag, or if you have a friend or loved one who really needs one for Christmas, be sure to check my Cafe Press Prime Factorization Store! 🙂

My posts on Mathematical Knitting and related topics are now gathered at Sonderknitting.