Mathematical – Page 10

Jan 25, 2016

Review of Numbed! by David Lubar

Award Winners, Children's Fiction Review, Fantasy, Mathematical 0

Numbed!

by David Lubar

Millbrook Press, 2013. 144 pages.
2015 Mathical Honor Book

I read this book while waiting for the Metro on the way to the National Book Festival – where I got to meet the author at the Mathical booth! I already knew I enjoy his sense of humor because of his Twitter posts as well as his writing, and I’m happy that he turned toward numbers with this book.

In Numbed!, the kids from Punished! get into new trouble at the Math Museum. They go into an experimental area where they’re not supposed to go, and an angry robot zaps them so they’re numbed. First they can’t do any math at all; when they fix that (by solving a problem in the matheteria, where a special “field” helps them), they can only do addition and subtraction, but not multiplication and division. When they fix that, they still can’t do word problems or apply mathematical reasoning to anything.

Now, as a math person, I really have to work hard at suspending disbelief for this story! Multiplication is repeated addition, so the idea that the kids would be able to add and subtract but not multiply didn’t work for me. Of course, the kids figured that out – that was how they got around the problem. But that areas of math are so distinct? No, I couldn’t quite handle that! And then the hand-waving involved in the robot being able to “numb” them and the matheteria having a “field” making it easier to do math problems? Aaugh!

But I really wanted to like the book. It won a Mathical Honor! And I like the author! So let’s point out all the good things about it. First, I do like the characters – boys who can’t stay out of trouble. At the start of the book, they don’t see what math is good for – and they definitely find out it’s good for many, many things when they lose the ability to do it.

I really enjoyed the high-level problems the boys had to solve to break their curse. The boys applied creative reasoning, and the problems and solutions were all explained clearly – and we believed that the boys could figure them out, at least in the enhanced “field.”

In general? The premise was a little hard for me to get past – but in practice, the book was a whole lot of fun. It’s also a quick read – I only read it while I was waiting for the Metro, not while the Metro was moving, and finished the whole thing on National Book Festival day.

Punished! has been very popular with kids in our county. I hope they’ll also find out about Numbed!. A silly school story – with math!

davidlubar.com
millbrookpress.com

Buy from Amazon.com

Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Fiction/numbed.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Source: This review is based on a library book from Fairfax County Public Library.

Disclaimer: I am a professional librarian, but I maintain my website and blogs on my own time. The views expressed are solely my own, and in no way represent the official views of my employer or of any committee or group of which I am part.

What did you think of this book?

Jan 16, 2016January 16, 2016

Zoe’s Prime Factorization Blanket!

Mathematical, Prime Factorization 0

Back in November, I finished my little niece Zoe’s Prime Factorization Blanket!

What is a Prime Factorization Blanket? Why, a blanket that shows the prime factorization of all the whole numbers up to 99, using a color for each prime number.

This is the same set-up as my niece Arianna’s Prime Factorization Blanket, as a matter of fact. But I used new colors for Zoe’s blanket, going with a lot of pink, because we already knew she was going to be a girl. (With Arianna, we found out she’d be a girl right when I got to the number 17, so in that blanket 17 is pink.)

The blankets don’t really need a pattern, but here are the specifications: I used Tahki Cotton Classic yarn, because it has so many shades available. Each square is a garter stitch square with 12 ridges and 12 stitches, which is easy to divide in 2, 3, 4, or 6 sections. For 5 sections, I did a plain row at the beginning and end. It’s done in entrelac, so you go across and knit the square for each number individually, then go back making the white squares, then do the next row of numbers, then a row of white. It’s much nicer than making the original sweater, because you can work on one number at a time, and don’t have to carry yarn across.

Here is Zoe’s Prime Factorization Blanket laid out flat (or sort of flat):

Here’s how it works. Starting in the bottom left corner (because graphs always have the origin in the bottom left), there’s a missing space for zero. Then 1 is pale pink, the background color:

2 was assigned the color pink.
3 was assigned the color red.

4 is our first composite number, 2 x 2. So I used two sections of pink. (If you look at the actual blanket, you can tell there are two sections, but it’s harder to tell in the picture.)
5 is prime, so it’s assigned a new color, yellow.
6 is composite, 2 x 3. So it gets a section of pink and a section of red.

7 is prime, so it gets a new color, purple.
8 is composite, 2 x 2 x 2. Three sections of pink.
9 = 3 x 3, so it gets two sections of red.

New row, so look back at the photo of the bottom right.
10 = 2 x 5, so it gets a section of pink and a section of yellow.
11 is prime, so it gets a new color, turquoise.
12 = 2 x 2 x 3, so two sections of pink and one section of red.
13 is prime, so it gets a new color, sea foam green.

Now the picture for the middle:
14 = 2 x 7, so pink and purple.
15 = 3 x 5, so red and yellow.
16 = 2 x 2 x 2 x 2, so four sections of pink.

Now the picture of the right side:
17 is prime, so it gets a new color, baby blue.
18 = 2 x 3 x 3, one section of pink, two sections of red.
19 is prime, so it gets a new color, olive green.

The next row starts at 20. The blanket goes all the way up to 99.

Here’s the top corner, so you can see some bigger numbers:

You can see the patterns nicely in the grid of the blankets. As an example of some simple patterns, the twos and fives line up in straight lines, but so do the elevens, in a diagonal line. There are lots more patterns which you can find the more you look at the blanket.

And Zoe likes it!

I’m gathering all my Mathematical Knitting links on my Sonderknitting page. (I hope to soon add coloring pages, too!) Check out the rest!

Dec 2, 2015September 11, 2021

An Outliers Scarf for Jade

Mathematical 2

I recently posted an explanation of my Probability Scarf, where I simply rolled a die to decide which of 6 colors to use for each row of the scarf.

But that represents a uniform distribution, where each color is equally likely — a little boring.

So I thought: Why not make a scarf using the normal distribution, a bell-shaped curve. I searched the web and found a site that would give me random numbers generated from a normal distribution.

I’ll use four colors:

Brown is for the center of the distribution (numbers within half a standard deviation from the mean). This is where most of the data will fall.

The next color has a bit more red in it, but it’s between red and brown. This will be for numbers between a half and one standard deviation from the mean.

The third color will be used for numbers more than one standard deviation from the mean, but less than one and a half standard deviation. It’s quite bright and red and pretty.

And finally — for the outliers — I bought a rainbow yarn. It turns out it changes colors very slowly, so you can’t necessarily tell that it’s rainbow-colored in the scarf, but it is bright and is slowly changing.

Also, about half the numbers are negative and half positive. I went with positive is for purl and negative is for knit.

And the point of the scarf? It is the outliers that make it beautiful! Yes, we need the nice middle-of-the-road, close to the mean folks — but the colorful ones are the outliers and add spice to life.

I’m planning to give the scarf to my daughter Jade, who has always been an outlier in several areas — and I fully believe that has a lot to do with why she is so wonderful.

The scarf is turning out lovely. I plan to continue until I run out of one color. (I bought two skeins of the brown yarn.) Yes, I am going to have lots of ends to sew in when I am done! I’m planning to do a crocheted edging in brown to cover up some of that.

I’m gathering all my Mathematical Knitting links on my Sonderknitting page.

Nov 27, 2015February 26, 2016

Fibonacci Swatchy

Mathematical 0

My sister-in-law is expecting a baby next June. Her toddler already has a Prime Factorization Blanket, and I just finished making a second one for a niece in another family. It’s time for something new!

Inspired by my Fibonacci Clock (not my idea, but a clock purchased via Kickstarter) and my Fibonacci Spiral Earrings, I’m thinking about making a Fibonacci Spiral Blanket.

The Fibonacci Sequence is simple. You start with 1, then each new number is the sum of the two numbers before it:

1
1 + 0 = 1
1 + 1 = 2
1 + 2 = 3
2 + 3 = 5
3 + 5 = 8
5 + 8 = 13
8 + 13 = 21
and so on. . .

I made a swatch to see if it would work, and I think it’s going to. Here’s the Fibonacci Swatchy:

It starts with the little white square, which represents 1. I planned to make the blanket 12 stitches by 12 garter ridges. I made the swatch 6 by 6, and think I may go with that for the blanket after all. The important thing is for it to be divisible by 3. It’s going to get big fast.

Okay, after the initial square, I picked up stitches along one edge of the square. I added a new color for this square, but it’s the same size as the first, still representing 1. Since 1 = 1 + 0, I used the first color (white), but added a new color representing the new entry in the sequence.

For the next square, representing 2, I picked up 12 stitches along both the previous squares. I use three colors — representing the two numbers whose sum in the new entry. This pattern will continue. Each new Fibonacci number will get a new color of its own — but I’ll alternate that with the two colors representing the two numbers I summed to get this number.

And in garter stitch it turned out very cool if you alternate rows of three colors — It turns out that you will have the yarn waiting for you when you’re ready to pick up that color again on the correct side. And the garter ridges work out to look like solid stripes. There are two colors in between the ridges, but because of the way the texture works, you see the matching color ridges together.

So in the swatch, the entry representing 2 was a 12 by 12 square alternating white, pink, and burgundy.

For the next entry, representing 3, I picked up stitches along the square I just finished plus one of the 1 squares, so that made 18 stitches, and I went for 18 rows. I dropped the first color white, and now alternated pink, burgundy, and a new color, lavender.

To finish it off, I chain stitched in a golden Fibonacci spiral. For the actual blanket, I’ll be a little more careful to make each curve circular.

I think this may make a fine blanket. The squares will get big quickly, so I’m not sure how far it will go. My brother and his wife should find out the baby’s gender in January. Though I’m thinking even if the baby is a girl, I may want to use more gender-neutral colors in the middle (these starting squares) and save pink for the bigger squares that will come later. But we’ll see. I also learned a little bit by swatching about how I want to pick up the stitches. But the main lesson is that alternating three colors in garter stitch works great! And crocheting on a golden spiral works great!

This is going to be fun!

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

Oct 14, 2015October 14, 2015

My Prime Factorization Hairnet

Mathematical, Prime Factorization 0

Our church is having a Stop Hunger Now Food Packaging Event next Sunday, October 18, 2015. As a form of publicity for the event, they’ve asked us to decorate a hairnet and take a selfie.

That was the moment I realized: I have a Prime Factorization Sweater, a Prime Factorization Cardigan, a Prime Factorization Scarf, a Prime Factorization T-Shirt, and have made Prime Factorization Blankets. But I didn’t have a Prime Factorization Hairnet!

Well, I soon remedied that!

Okay, it’s not knitting. But I printed a chart I’d made of numbers color-coded with their prime factorization for the Prime Factorization T-shirt. Then I simply cut out the individual squares and glued them to the hairnet in a spiral pattern. So it goes from 1 to 100.

How it works? Each prime number gets a new color. Composite numbers are divided into sections with a section for each factor. Each section is colored according to that prime’s color. For example, 42 = 2 x 3 x 7, so the square for 42 is divided into three sections, colored blue for 2, red for 3, and green for 7.

This selfie not only shows the Prime Factorization Hairnet, it also gives a glimpse of infinity!

Oh, and I’m gathering all my Mathematical Knitting (and other mathematical creations) at Sonderknitting. Eventually, I’ll add mathematical explanations and patterns and activities and other good things.

I can safely say that mine is the most educational hairnet selfie posted yet!

Oct 7, 2015February 27, 2016

My Probability Scarf

Mathematical 2

I’ve started collecting my Mathematical Knitting posts at Sonderknitting, a Mathematical Knitting Gallery.

But I’d never done a post about my Probability Scarf.

This is not my idea. I don’t remember where I saw the instructions, but they are easy and a lot of fun.

1. Choose six colors of yarn that go together well. Assign them numbers from 1 to 6.

I chose leftovers from my Prime Factorization Sweater.

2. You’ll be knitting a scarf the long way, using the ends as fringe. Start by casting on to a circular needle however long you want your scarf to be. (Try to keep it loose!)

3. For each row, roll a die to decide which color to use. Flip a coin to decide whether to knit or purl.

4. Continue in this manner until you’ve run out of one of the colors.

You now have a scarf demonstrating the Uniform Distribution.

This scarf was fun to knit. It was hard to stop knitting, because I kept wondering what the next row would look like.

It occurs to me that it would be fun to do a Probability Scarf using a different probability distribution. You could find a generator based on another distribution (where the colors wouldn’t all be evenly distributed) and use that to decide which color to use. This would be fun if you wanted to use a second or third color just for highlights. Or maybe you didn’t have the same amount of each yarn. Maybe that will be a future project….

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

Sep 14, 2015

Review of I See a Pattern Here, by Bruce Goldstone

Children's Nonfiction Review, Mathematical, Starred Review 0

I See a Pattern Here

by Bruce Goldstone

Henry Holt and Company, New York, 2015. 32 pages.
Starred Review

I love Bruce Goldstone’s books about math concepts. They are bright and colorful and draw kids in – and explain the math concepts in simple language, with helpful, dramatic visuals.

This one is about patterns. He explains them using simple language and has a little box giving the mathematical vocabulary where it’s appropriate. As in his other books, he starts simply and builds.

The book covers repeating patterns, then translations (“slides”), rotations (“turns”), reflections (“flips”), symmetry (“equal sides”), scaling (“changing sizes”), and tessellations (“tile patterns”). The many, many varied pictures make the concepts so clear.

For example, he uses photos of quilt blocks, tiles in the Alhambra, kaleidoscope images, lace patterns, tire treads, animals, architecture, beads, stamped patterns, and a 2000-year-old Peruvian cloak.

This is a beautiful book that will get kids noticing the patterns around them and give them a new vocabulary for talking about those patterns.

brucegoldstone.com
mackids.com

Buy from Amazon.com

Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/i_see_a_pattern_here.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Source: This review is based on a library book from Fairfax County Public Library.

What did you think of this book?

Jun 1, 2015

Review of If . . . by David J. Smith

Children's Nonfiction Review, Mathematical 0

If . . .

A Mind-Bending New Way of Looking at Big Ideas and Numbers

by David J. Smith
illustrated by Steve Adams

Kids Can Press, 2014. 40 pages.

The author of the brilliant If America Were a Village is at it again, using scale to give children a feeling for enormous numbers. Here’s what he says on the first page:

How big is Earth or the Solar System or the Milky Way galaxy? How old is our planet and when did the first animals and people appear on it? Some things are so huge or so old that it’s hard to wrap your mind around them. But what if we took these big, hard-to-imagine objects and events and compared them to things we can see, feel and touch? Instantly, we’d see our world in a whole new way. That’s what this book is about – it scales down, or shrinks, huge events, spaces and times to something we can understand. If you’ve had a doll or a model airplane, you know what scaling down means. A scale model is a small version of a large thing. Every part is reduced equally, so that you don’t end up with a doll with enormous feet or a model plane with giant wings. And when we scale down some really huge things – such as the Solar System or all of human history – some of the results are quite surprising, as you are about to see…

The book goes on to look at such scenarios as:

If the Milky Way galaxy were shrunk to the size of a dinner plate…
If the planets in the Solar System were shrunk to the size of balls and Earth were the size of a baseball…
If the history of the last 3000 years were condensed into one month…
If the inventions of the last 1000 years were laid out along this ruler…
If all the water on earth were represented by 100 glasses…
If all the wealth in the world were represented by a pile of 100 coins…
If average life expectancy (the number of years people live) were represented by footprints in the sand…
If today’s world population of over 7 billion were represented by a village of 100 people…
If your whole life could be shown as a jumbo pizza, divided into 12 slices…

With each scenario, graphics on a double-page spread show how the hypothetical object would be divided up, with some surprising results.

In the your-life-as-a-pizza example, 4 of the 12 slices would be work and school and 4 of the 12 slices would represent sleeping. In the wealth example, we see one person standing on top of a pile of 40 coins, 9 people on top of the next 45 coins, on down to 50 people standing on the one lone last coin. With footprints in the sand, we see the footprints from some continents don’t go nearly as far as those from others.

The population example may be the most interesting, because the author goes back in time. If today’s population were represented by a village of 100 people, the village in 1900 would only have 32 people, in 1500 only 8 people, and in 1000 BCE, there would have only been 1 person.

kidscanpress.com

You get the idea: These ideas and images give you a grasp of the large proportions between these things and a handle for understanding enormous numbers.

Buy from Amazon.com

Find this review on Sonderbooks at: www.sonderbooks.com/Childrens_Nonfiction/if.html

Disclosure: I am an Amazon Affiliate, and will earn a small percentage if you order a book on Amazon after clicking through from my site.

Source: This review is based on a library book from Fairfax County Public Library.

What did you think of this book?

May 19, 2015February 8, 2016

Pascal’s Triangle Shawl #2

Mathematical, Prime Factorization 3

Hooray! Hooray! Today I finished my second, prettier Pascal’s Triangle Shawl!

Pascal’s Triangle is the triangle with 1s on the edges, where each entry is the sum of the two entries above it.

So the beginning rows work like this:

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

Now, what I did was choose a color of yarn for each prime. Then each entry in the triangle is factored, and each number is shown by the colors of its factors.

I did the same thing with my first Pascal’s Triangle Shawl. With this one, since there are only the primes 2, 3, 5, 7, 11, and 13, I decided to use progressively darker shades of pink and purple, so the shawl would gradually get darker.

Here is a closer look at a section of the shawl:

This next picture shows that along the second row, we have the numbers simply in sequence.

For math nuts, each row also contains the binomial coefficients, the coefficients in the expansion of
(a+b)^n

This means that the rth entry in the nth row can be calculated with the formula:
n!/(n-r)! (Counting the entries in each row as 0 through n.)

Some examples: The 2nd entry in the 5th row is (5×4)/(2×1) = 10

The 3rd entry in the 7th row is (7x6x5)/(3x2x1) = 35

Now, I factor all the numbers in my shawl, so for big numbers, it doesn’t matter what the actual number is, but the factorization is easy from the formula.

For example, the 4th entry in the 15th row is (15x14x13x12)/(4x3x2x1) = 3x5x7x13

You can see some of the bigger numbers in this picture:

Now, there are a couple of characteristics which I believe make the shawl especially beautiful.

One is that because these are the binomial coefficients, once you get to the row of a prime number, every entry in that row has the prime for a factor.

This is easier to see with the actual shawl in front of you, but here again is the big picture. You can see that once a new color starts, it goes all the way across the row.

What’s more, by the distributive law, since every entry in a prime row has that prime as a factor, all the sums of those numbers will also have the prime for a factor — and we end up having inverse triangles of each color.

Here’s some more detail:

Of course, the very coolest thing about it is that, even if you have no idea of the math involved, the combination is beautiful.

And that simply makes me happy.

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

Apr 23, 2015April 23, 2015

Crazy 8s Math Club and Living Venn Diagrams

Events, Mathematical 0

This week I brought my camera to Crazy 8s Math Club! We were learning about Sets and Venn Diagrams – and look at those faces!

Crazy 8s is a Math Club sponsored by BedtimeMath.org. They provide the ideas and materials, and the library provides the place.

Here is a set of kids with brown eyes. We had a Flat Visitor from California who also had brown eyes!

When we started the 3-set Venn diagram, I thought they could start with cars and trucks. They caught on quickly!

And finally, a living Venn diagram. The kids figured out where they belonged depending on whether they had brown eyes or not, whether they could curl their tongue or not, and whether they fold their hands with their left or right thumb on top. I’m happy to say that the kids who didn’t fit in any of those sets were excited to be “in the universe.”

And afterward — some silliness with the glowsticks (which they got to take home).

More proof that Math is Fun — and kids know it!