# Category Archives: Resources

## Take a break from the workbooks

Whether you have a student who lives and breathes math or a reluctant one who balks at the idea of worksheets, living math books, where math is integrated into a story, might be the perfect tool for extra math learning.  Unfortunately many story based math books are low quality in the sense that the math itself is either very basic or the storyline is contrived and uninteresting. The following resources have met my criteria for both covering math exceptionally well and also being something that a child might be motivated to pick up and read over-and-over again on their own.

Sir Cumference

This is a wonderful series of picture books by Cindy Neuschwander for the youngest of students. The stories follow the adventures of the Knight Sir Cumference of the Round Table, his wife, the Lady Di of Ameter, and their son Radius as they uncover geometric properties of various shapes and other math truths. Books in the series:

Penrose the Mathematical Cat

The Adventures of Penrose the Mathematical Cat by Theoni Pappas is by far our favorite living math book and great for kids aged 6-12, especially ones that enjoy animals and unusual math topics. When he was younger my son went through a phase of liking books that explore the perspective of animals trying to understand the human world and this one definitely fits into that category while also having an incredible amount of interesting math layered into it. The story part of it is a bit thin, but Penrose the Cat is such a fun character that you don’t notice too much. Topics include things that normally fall well outside of a typical elementary math curriculum including chapters on binary numbers, Fibonacci sequence, tangrams, and infinity. The sequels –  The Further Adventures of Penrose and Puzzles from Penrose  are also worth getting if the first book is a hit with your student.

The Number Devil

The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger is about a boy visited by a math loving demon in his dreams. Covers both basic and advanced number concepts in a casual way with a highly integrated and whimsical story. The chapters on Fibonacci numbers and Pascals’ triangle are particularly well done. Perfect for ages 8 and up.

Murderous Maths

The Murderous Maths series by Kjartan Poskitt is our latest acquisition of living math books and I’m glad I splurged on the box set, because its been a huge hit! Each book covers a different topic and is a dense chapter book embedded with comics, diagrams, and boatloads of humor. The topics range widely from everyday math basics like geometry, measurement, and probability to quick mental math tricks to math trivia to far out there subjects like graph theory. The humor and presentation make it perfect for math lovers and not-so-mathy students alike, especially for those in the 9-12 year-old range. This series is published in the UK but easy to find on Amazon at a reasonable price and a few are also available in Kindle format.

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## Math roundup this week!

I spend a lot of time trying to keep math interest high by introducing concepts in a fun way, exploring old material from new angles, and bringing hands-on learning to our “Math Lab” whenever possible. This isn’t always easy when delving into more advanced math topics – baking with fractions is one thing but information theory, combinatorics, and algebra are quite another. However, I firmly believe this is the only way to teach talented math students at the elementary level without burning them out or turning them into grinds. Fortunately I love math and computer science so the hours I spend researching and planning activities is never a chore. It does involve a lot of experimentation and dabbling though. Not every topic is brought to completion in a neat, well thought out package. A lot of math enrichment is about throwing cool ideas out there and seeing what sticks. My hope is that even if something is not a hit, I’m planting a seed for future investigations. This week I’m giving you a peek behind the curtain to see how random our weekly math explorations really are sometimes. Fortunately most of them were a hit, but that’s not always the case!

### Flexagons

If you don’t know what a Flexagon is, you are really missing out, and even if you’ve heard of them you may not be aware of all of these awesome mathematical properties. First discovered by British graduate student Arthur Stone in 1929, Flexagons are geometric models that have hidden faces when flexed in certain ways. They are super simple to make out of paper and a fun project for home or the classroom. I was pretty dismissive of the math behind them until I started doing some research and then was blown away by their awesomeness. The famous physicist Richard Feyman even invented visual state diagrams, which are especially complex for hexa-hexaflexagons and my student has had a great time trying to make his own.

We’ve been using two new resources this week to study probability in a novel way. The Amazing Mathematical Amusement Arcade by Brian Bolt is a fantastic book that distills famous math puzzles into a style that is more visually appealing for kids. I picked this up at the library last week, and just coincidentally a puzzle in this book called the Gambler’s Secret Strategy completely mirrored a more technical chapter I was reading on non-transitive dice paradoxes in Martin Gardner’s Colossal Book of Mathematics. The crux of the puzzle involves dice that have equal total values, but different numbers on each face, such that they allow a gambler to always have a better chance of winning in a head-to-head roll even when letting his opponent choose first.  We created paper dice and built conditional probability trees to prove that the gambler always has the advantage. Definitely a topic we will be revisiting.

### Pentagonal Numbers

I put this sequence on the board one day and told my student to ponder it.

1, 5, 12, 22, 35, 51, 70, 92, 117…

The first day he told me that he didn’t know what the pattern meant but that he had figured what the next number in the sequence was (145). He then created a formula for figuring out the next number in the sequence given the two previous. Then, since he had trouble visualizing what these numbers meant, we drew pictures to illustrate why they are called pentagonal numbers. Now we’re trying to figure out what the connection is between this series and the series of triangular numbers. After that we will explore the set of integers that cannot be expressed by a sum of three pentagonal numbers. This one has been a huge hit. I love these long term math explorations!

I have a series planned on recreational math books for kids. Our latest acquisition in this category is a British series called Murderous Maths. Surprisingly dense and a whole lot of fun. Expect a detailed review soon.

Beast Academy 5c. The long awaited next book in this elementary math series from Art of Problem Solving. My student has moved beyond the math in this series but he still loves reading the comic book adventures of the little monsters as they solve math problems, and the workbook makes for great review.

The Thrilling Adventures of Lovelace and Babbage – a fun and fascinating romp about two of the founders of the computer age. Only slightly educational but a great graphic novel for any geek. I wasn’t expecting my son to get into, but he’s been dragging it around and re-reading like crazy.

### Logic

We’ve also been taking our digital logic to the next level with a free circuit design tool called Logism. It’s very easy to use and allows for quick design and simulation of functional circuits. He had no problem completing simple projects of building a half-adder and full-adder, and is now working on a 7 segment display controller. So many possibilities here and an easy way to explore electronics without an expensive science kit.

That’s our math roundup for the week! We’ve been trying to enjoy our gorgeous fall weather here on the east coast by doing less school and spending more time outdoors, but we always have time to squeeze in lots of math so stay tuned.

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## Stretching math through the summer (part 1)

Summer is still a few months away but I know many parents are already looking for ways to build or maintain their child’s math learning over the summer. There are several great options that I’ll be discussing in the coming weeks for every age and ability level, but for math-talented students in grades 3-5 my first recommendation is usually Beast Academy from the Art of Problem Solving (AoPS).  Although Beast Academy is a full year-long comprehensive math curriculum, it comes in an incredibly fun and cute package, which makes it ideal for summer.

Every grade level has 4 parts (A-D) each consisting of a comic book, which serves as the textbook, and also a corresponding workbook.  The Beast Academy characters in the comic books are fun and talk the student through new concepts, while the workbook problems themselves are very puzzle and game-like. True confession – my son has been through the entire available Beast Academy program and still likes to use the comic books for bathroom reading. They are that engaging!

My only word of warning is that Beast Academy is both challenging and unconventional. Art of Problem Solving made a name for itself with advanced high school and math contest prep materials, and Beast Academy carries their tradition of using a discovery based approach to mathematics and deep problem solving to the elementary level. It is ideal for students who love math and love to figure things out for themselves, but I would not recommend the workbook for a student who already struggles with math, as the problems found in the book can sometimes be intimidating.

When using as a summer supplement expect to get through only one book, unless you’re using it a grade or two behind as review, something that I actually recommend as most fifth graders who have been taught conventionally could still get a lot out of the lower grade Beast Academy books. The depth of this series is incredible – in addition to standard fare like multiplication, fractions, and decimals, Beast Academy covers some topics that are seldom seen in elementary school such as logic and basic combinatorics.   At the same time I also wouldn’t hesitate to try these materials with a very advanced math student who is a grade or two younger (my son started 3A in first grade). The fun presentation and packaging make it ideal for a student who is intellectually ahead but lacking the maturity to jump into more serious math texts.

AoPS has full sample pages of both the guidebooks and workbooks for each level on their website.  Materials can be purchased directly from AoPS or through some discount homeschool catalogs like Rainbow Resource.

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## Starting after school enrichment with math-talented students

If you have an elementary student who is talented in math (or just simply loves it), figuring out how to supplement their math education can be tricky. You may sense that they need more than what school is providing, but not know where to begin. With remedial students it’s usually clear which skills need to be shored up or which gaps in conceptual thinking need to be bridged. However, for a student who is great at math and is under-challenged in school, the answer is less obvious. Doing more of the same is rarely the answer. Math-talented students don’t need more work, they need different work; work that really stretches their brain, but not so tough that it’s discouraging.

Acceleration is usually the first option considered, and it can be part of the solution, but its not always the perfect choice. Frankly, most math programs don’t actually get much harder as they go along. They introduce new topics and procedures but rarely delve into the kind of multi-step problem solving or open ended questions that will really develop math ability.

Also getting too far ahead in the school curriculum can be a real concern for some parents. This isn’t a problem if you’re a homeschooler, have a very flexible school that will work with your child’s acceleration, or are willing to continue supplementing at home until high school, but otherwise there can be consequences to working too far ahead and having a child bored in the classroom.

Fortunately it doesn’t have to be an either/or decision. There are many topics in mathematics that are both broader and deeper than basic arithmetic and that go way beyond the standard pre-algebra-to-calculus pipeline. I’ll be talking more about this in future posts. However, my first recommendation for any student, of any age, is to improve their problem solving skills.

An easy way to get started is with the Singapore Challenging Word Problems (CWP) series. A great first step is to simply buy the CWP book that is at a level appropriate for your child, tear out a page, grab a magnet, and put it on the fridge. Make this “The Problem of the Day.” Do it before school or after school or whatever time of day is best for your family. Perfectionism among gifted students is common, so I believe it’s important to keep it fun and not get caught up in whether your child gets it right or wrong. Doing math outside of school should not be a chore, but about learning to love problem solving.

There are other great problem solving books out there, but this is the one that I’m most familiar with. The problems in CWP have incredible depth and are extremely well written. The best thing about this is that you’re hitting a lot of different skills in one time-efficient package. You get multi-step problem solving, a gentle introduction to algebraic thinking, and basic arithmetic practice on a variety of topics. My only caveat is that CWP, as the name says, is challenging, especially if you haven’t had experience with Singapore math, so I often recommend buying a grade level down to build confidence and patience.

The reason I like the “problem a day” approach is that it fits in with my two basic educational principles.  The first basic principle is that starting small is the best way to begin. There are a few people who can radically overhaul their lives in a small amount of time, but its rare. For most parents, finding the time and energy to squeeze in some math enrichment between sports and music and homework and dinner and bedtime is tough. Starting small means you will actually do it.

This leads into my second basic principle, which is that a little bit every day is better than a lot all at once. With daily work you have higher retention and the extra minutes quickly add up. I know the idea of starting with 10 minutes a day of extra math sounds like the too-good-to-be-true ab workout on the cover of a fitness magazine in a supermarket checkout. However, that extra problem a day can painlessly lead to an extra hour of math per week, and if you keep it up all year round you’re potentially talking about the equivalent of two full grade levels in the course of an elementary education.

So if you’re not sure where to start with your math-talented student start simple with a daily word problem.