# Focus on Math

## Helping children become mathematicians!

### Jayden’s Rescue (Revisited)January 22, 2014

We were also prepared to do the mathematical questions that would arise in the story as part of the rescue task.  The first problem that comes in the book is this:

I am the father of nine sons, all one-eyed monster boys.

I keep an eye on all my lads as they play with their toys.

A three-eyed monster once dropped in and brought his sons along.

Three bulging eyes were on each guest.

Oh! What a blinking throng!

Together all the monsters had exactly forty eyes.

How many three-eyed kids were there?

The numbers tell no lies.                              Jayden’s Rescue, p. 24

It took a bit longer than I had anticipated getting to that question on p. 24 (not to mention that that the lunch period began 15 earlier than I had expected it to!) so the students ended up not having a chance in the morning to share their solutions/strategies to the question. I left that in Mrs. DeGroot’s capable hands to finish up.

I believe the book is presently out of print, but copies are available from used-book sellers. It is definitely a book worth tracking down and reading to a class in the grade 4 to 6 range.

Mathematically yours,

Carollee

### 12 Days of Christmas: Canadian StyleDecember 11, 2013

If you are looking for a great Christmas-themed book for the young (or young at heart), you will want to check out A Porcupine in a Pine Tree: A Canadian 12 Days of Christmas (written by Helaine Becker, illustrated by Werner Zimmerman). Of course, the math connection is in the counting and in the adding up of all the items being given during the 12 days – all totaled it adds up to quite a tidy sum.

Certainly, any book with numbers tends to make me smile, but this one more than most. It is seriously delightful! It offers Mounties frolicking, squirrels curling, moose calling, hockey players a-leaping, and more.

The hardcover version is available from Amazon.ca (not Amazon.com) for the bargain price of \$12.26 (price valid as of date of posting). Actually, for just a bit more, you can get the gift set version with the hardcover book plus an adorable little plush porcupine (sounds like an oxymoron doesn’t it!)

I am not affiliated with the author or publisher – I was given a copy of this book last year for Chrismas and I love it! I know you will, too.

May your days be merry and bright!

Mathematically yours,

Carollee

### Hat Tricks: Logical Thinking with ShadowchildMay 21, 2013

Anno’s Hat Tricks (by Akihiro Nozaki and Mitsumasa Anno) is a wonderful way to introduce children to the realm of logic and the powerful word “if”. The book goes through a series of “tricks”, all of which can be solved by applying that “mathemagical” word “if”– a word that opens doors to new ideas. Children are introduced to the concept of using “if” statements to test the truth of an idea or supposition in a logical way. The reasoning pattern of “if…then” can be very useful, and, indeed, branches of modern mathematics have been developed by applying the word “if”.

This delightful book is mainly about three children: Hannah, Tom (both of whom are clearly seen) and Shadowchild (who exists on the page only as a shadow). The writer gives a series of scenarios in which the reader is shown a certain number of hats (all either red or white) which are available, and then which ones of those hats are being worn by Tom and Hannah. We are to use logic to deduce what colour hat Shadowchild is wearing. Although the first number of scenarios in the story are quite easy, the difficulty level increases throughout the book, with the final trick being the most difficult. (If you are not sure of your own level of logical thinking, there are several pages at the end of the book devoted to parents and other older readers that will offer some assistance in the logic being applied in the different tricks.)

Sadly, I think the book is no longer in print, but it is well-worth your while to track down a copy. I know and your children will enjoy the challenges presented.
Mathematically yours,
Carollee

### Bean Thirteen — a Lesson in “Fair-Sharing”June 15, 2012

One of my recent purchases was the delightful book Bean Thirteen by Matthew McElligott. The story tells the tale of Ralph and Flora, two bugs who were picking beans for dinner. Ralph warns Flora not to pick the thirteenth been (as it is unlucky!) but as Flora does not agree with Ralph, she goes ahead and picks bean the thirteenth bean. Then dilemma begins. If they split the beans between them, bean 13 is left over. If they invite one, two, or four friends over to share the beans with, bean 13 is still left over. (And there was real trouble when they only tried to invite 3 friends over!) The final solution for the problem is a real-life example of problem solving at its best. Bean Thirteen is a great book to use for developing the concept of division for K-3 students.

After reading the book to my two grade 2 classes, I had them do an activity based on the book. Students each counted out 18 bingo chips to represent beans, and then they were asked to share them on “plates” for different numbers of friends. I provided cut up pieces of construction paper to represent the plates, and the students shared out the beans equally on the plates. (We called this “fair-sharing”.) Beside the number of plates (or number friends sharing) listed on the recording sheet, students wrote down the number of beans on each plate and the number of beans (if any) left over.

Because of the concrete nature of the activity, everyone was able to be successful. Early finishers were allowed to take a different number of “starting” beans and explore what would happen when that number of beans was shared.

If your school library does not have a copy of this book, ask your librarian to order one in. It is well worth having.

Mathematically yours,
Carollee

### Books! Books! Books! The Literature Connection (part 2)February 7, 2012

It seemed appropriate to follow up the last post with some more of my favourites. The books in these two posts are the ones I use over and over again when I go into different classrooms. I have many more books in my collection, but these are the “go-to” books that I keep reaching for.

Every Minute on Earth by Steve Murrie and Matthew Murrie is a great book to use with students in the grade 4-9 range. This book is chock full of interesting facts about the earth, space, the human body, technology, animals. food, pop culture, and sports. Many pages use many big numbers to tell about the event. For instance, on the page telling that more than 34,000 plant species are threatened with extinction each minute, the authors also tell the readers that seeds for more than 6 million different plant species are stored in 1,300 sites around the world. Of those being stored, about 15% are seeds for wild plant species. There are so many questions that can be generated from the numbers from this one-page story. Sometimes I ask the questions, but often I ask the students to generate (and solve) questions from the information.

Bat Jamboree by Kathi Applet can be used when you want students to explore adding sums of consecutive numbers. The story tells about the number of bats in different groups that are performing at the jamboree, starting with 1, then 2, … up to 10 bats. At the end all of the bats make a pyramid, and I always stop reading so students can figure out ways to calculate the number of bats in the pyramid. Elementary students do not generally come up with the sophisticated algebraic formula that some of you may have encountered [f(n) = n(n+1)/2], but it is rather amazing what patterns they can find.

How Many Feet in the Bed by Diane Johnston Hamm is a wonderful book for using with primary children. In the story a family of five (mother, father, young daughter, young son, and baby) get in and out of the parents’ bed in the course of a morning. It is a counting by two book on that level, and primary children tend to enjoy this. I have followed it by asking children to draw their family gathered in one bed and and telling how many feet. I have also given grade 2-3 children a scenario of a family with dogs and cats as well as children. Then, if I tell them there are 12 feet in the bed, I have them find different combinations of people and animals that make the requisite number of feet. It becomes a great patterning question if the people, animals, and feet are recorded on a chart.

I bought 365 Penguins by Jean-Luc Fromental and Joelle Jolivet sight unseen. I was intrigued by the name, especially having grown up with a sister who was a penguin fanatic (actually, she still is!). I figured any book about penguins and numbers would have to be fun, and I was not disappointed. The premise of the story is that a family is being anonymously sent 1 penguin each day for a year, an oh! the numbers that are generated to play with. There are numbers about pounds of fish, cost to buy the fish, and storing the penguins, just to name a few. Questions can easily be created for addition, subtraction, multiplication, and division. Students love the book — and so do I!! This one is a must-have. (Incidentally, the story was originally written in French, so it would be great to track it down in that language for any French Immersion classes.)

The 512 Ants on Sullivan Street by Carol A. Losi is a rhyme about ants carrying off the food set out at a picnic. It is basically a book about doubling. At the back of the book are some suggestions for activities written by Marilyn Burns. One Grain of Rice by Demi is an Indian folktale dealing with this same principle of doubling numbers. I have used both in the classroom many times.

Although most elementary and middle school curricula do not specifically include a logic component, I am fascinated by how well Akihiro Nozaki and Mitsumasa Anno handle the topic in their book Anno’s Hat Tricks. The reader is part of the book, called “Shadowchild” (since the shadow always can be seen on the page). Two children are in the story, always wearing hats. We are told how many hats there are (always in either red or white) and we can see the hats on the two children. From what we see, we, as Shadowchild, must deduce what colour of hat is upon our own head. (We can, of course, see in the shadow that we are indeed wearing one, but we cannot tell its colour in the shadow.) Each time the hats are changed the challenge level goes up. It is a great book to use with older elementary and middle school students.

I hope you have the opportunity to use one or more of these books in your classroom. I know your students will enjoy the literature connection, and I’m betting you will, too!

Mathematically yours,
Carollee

### Books! Books! Books! The Literature ConnectionFebruary 3, 2012

Recently the Literacy Support Teacher in our district and I (the Numeracy Support Teacher) joined forces and did a workshop for teachers. Much of what we did that day connected literacy and mathematics through the use of picture books. I am sure that many of you are aware of some great books to use in math lessons, but I shall share some of my favourites here (in no particular order).

Marilyn Burns’ book Spaghetti and Meatballs for All is a classic and can be used in a number of ways in the classroom. It is particularly useful for playing with the concept of perimeter as tables are rearranged to accommodate guests coming for dinner.

Amanda Bean’s Amazing Dream by Cindy Neuschwander is a charming story about a girl who loves to count, but learns that multiplying is a much faster way to count things in rows or groups. I have read this book to many different classes and followed the reading by a multiplication word problem, usually at a “challenge” level for the students. They multiply by using what they know about numbers (breaking them apart, using repeated addition, etc.) to build an understanding of multiplication.

One Hundred Hungry Ants by Elinor J. Pinczes is another classic. It is, in fact, one of the first literature books I ever used in my classroom years ago. It makes use of factors of 100 to divide the ants into even rows, and is a great introduction to factoring.

The Right Number of Elephants by Jeff Sheppard should not be missed. It uses rather fanciful situations and describes the right number of elephants needed to fit the bill. For instance, “When you go to the beach with all of your friends on a very warm day and you simply must have shade, the right number of elephants is 8.” I love reading this book to young children (K and/or grade 1) and then following the story with an activity in which each student thinks of a situation and the right number of “something” for that. For instance, children have written and illustrated these: “the right number of pets is 3” (since she had 3 pets); “the right number of pencils in a pencil box is 5”; “the right number of shoes is 2”. It is a great way to talk about numbers all around us.

Measuring Penny by Loreen Leedy is the perfect book to use to introduce a measurement unit. In the book, Lisa’s homework assignment is to measure something, and she chooses to measure her dog, Penny. The wonderful thing about this task is that there are so many attributes to measure and thus many units of measurement are explored (standard and non-standard, metric and imperial). Students will want to go home and measure their own pets!

If your students are learning about angles, circles, and circumference, you will want to read them The Librarian who Measured the Earth by Kathryn Lasky. This book tells the story of how Eratosthenes calculated the circumference of the earth more than 2000 years ago. His calculation was correct within 200 miles, or 99.2% accurate — pretty good for being done in a very “low-tech” manner, don’t you think?

These are only a few of the books I love using with students. I’ll post more of my favourites later. In the meantime, at the side of the blog you will find an “interesting link” which will take you to a site filled with scores of possibilities for using literature in your math class. It is definitely worth your time to explore the list there.

Mathematically yours,
Carollee

### Fibonacci Numbers: A Fascinating SequenceJanuary 10, 2012

I was recently given the gift of this delightful interactive book written by Emily Gravett. Although it appears to be a children’s book, The Rabbit Problem can be appreciated on the adult level as well. This tale of Lonely Rabbit and Chalk Rabbit is actually a retelling of a scenario that, according to Wikipedia, first appeared in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci. Fibonacci considered the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
•    At the end of the first month, they mate, but there is still only 1 pair.
•    At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
•    At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
•    At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The question, of course, is how do we know how many pairs of rabbits there will be at the end of any given month??***
The answer lies in the Fibonacci sequence of numbers, a fascinating set of numbers that keep popping up in nature in amazing ways. The sequence begins with 0 (or should I say “can begin”?), then add 1, and from there the next number in the sequence is always derived from adding the two previous numbers. So the third number is 0 + 1 or another 1, the fourth number is 1 + 1 or 2, then 1 + 2 or 3, then 2 + 3 or 5 and so forth, giving this sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 …
I recommend this short  video http://youtu.be/ahXIMUkSXX0 by Vi Hart to jumpstart your thinking about these numbers (it’s about 6 minutes long). She builds the topic of Fibonacci numbers off the topic of spirals, so be patient and the number part will come. As you watch, keep in mind the kinds of explorations that you and your class can pursue with pinecones, pineapples, artichokes, cactus fruit, flowers, and such. Often the number of seeds that show up in fruit and vegetables is a Fibonacci number. Try counting the seeds of the next apple or orange you eat!
This website http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html has a whole host of information about connections to the Fibonacci series. Note especially the section about plants. You might be able to do some of your explorations from the photos included here is real fruit, pine cones, etc are not readily available.
Exploring Fibonacci numbers can be a great “hook” to grab students’ interest about numbers and mathematics.
Mathematically yours,
Carollee
***By the way, the answer the the rabbit question is this: at the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (which is the number of pairs in month n − 2) plus the number of pairs alive last month (n − 1). This is the nth Fibonacci number.