Welcome to MathBait™ Multiplication Part 5! In some ways, this was the post that inspired it all. Having seen the power of Napier's Bones we wanted to highlight its usefulness in teaching multiplication. From this seed sprouted a fruitful tree with dozens of activities, lessons, and games to help your students master multiplication and have so much fun in the progress.
Thus far in the series students have grown from skip counting to dual processing to building fluency. Now it's time to continue on our previous work while also introducing students to multi-digit multiplication. We aren't talking 2-digit by 1-digit or basic values. In the next two parts of MathBait™ Multiplication, your students will be multiplying very large numbers with ease. How? By playing of course!
In this article, we explore a very old method of multiplying that every student should be introduced to. We'll show why playing with sticks can dramatically help strengthen student multiplication, show you how to make your own, and provide a fun digital version to engage with.
“Playing with sticks can dramatically help strengthen student multiplication skills”
Click on a drop down to explore each section below. We recommend bookmarking this page for easy access to each activity.
What is Napier's Bones?
In the 1600's, calculators were not commonplace. Worse, many did not have access to education. Yet, like today, math was everywhere. Thus, there was a need for ordinary people to be able to complete computations. Enter John Napier.
Napier was a Scottish landowner who saw math as a hobby. He was devoted to taking care of the Gartness estate and used his talents as an inventor to excel at his duties. Napier applied math everywhere, including to improve the land by applying scientific thought to agriculture.
While most famous for his invention of the logarithm, Napier had many extraordinary contributions including decimal notation for fractions, and what we are here for: Napier's Bones.
Solving the problem of allowing anyone to compute easily, Napier's Bones is often considered the first practical calculator. When you think about the first calculator, you may automatically imagine a large and complex device, but surprisingly, Napier's Bones is easy to make and fit in the palm of your hand.
Invented in 1615, Napier used his brilliant mathematical mind along with his talents as an inventor to construct a remarkably insightful and easy-to-use device that can multiply, divide, and even find square roots quite simply.
This makes Napier's Bones ideal for young students. Not only are the bones fun to play with, students need only the ability to add and subtract to harness the power of the device.
So how does Napier's Bones work? Consider the standard multiplication table. On its face, it is a 10-by-10 square (our table below adds an additional "0" column) constructed by skip counting that allows us to reference the product of two values with factors of 10 or less. But the multiplication table is so much more! The periodic table of numbers, every bit of our multiplication table is hiding secrets, connections, and beautiful relationships between values.
Recall our activity Table Mash Up from MathBait™ Multiplication Part 3. In this lesson, students explored the multiplication table from different perspectives. This is just what John Napier did. Using his deep understanding of mathematics, he broke apart the table and began rearranging its columns.
Taking the columns for 1, 4, and 6, Napier could now place these next to each other and examine their relationship. If he focused on the third row he noticed that each value was exactly the third multiple of his column headers (in this case 1×3, 4×3, and 6×3). The resulting number was 31218 which had no practical use, but Napier had a deep understanding of mathematics. If he wanted to multiply 146×3, the standard steps would be to find 3×6, 3×4, and 3×1, exactly the values that lay in front of him.
Taking into account place value, the number 31218 could be thought of as (31)(21)(8) where each value in parentheses corresponds to the place value the number would be applied to. Adding together the numbers in each place value led to (3+1)(2+1)(8)=(4)(3)(8)=438 (note the juxtaposition used here refers to place value rather than multiplication).
Grab your calculator! The value of 146×3 is precisely 438.
146×3=438
Amazing! Napier's rearrangement of the multiplication table is the magic behind his bones.
John Napier constructed a powerful manipulative. Each rod consists of the multiples of each digit and thus, in essence, are simply columns of the multiplication table. Rather than jotting down each product, Napier split each cell into two places: ten's and one's. This allowed his rods, or "bones", to create joints, and rather than carrying over as in the standard algorithm, the joints are already situated so that each place value lines up for easy addition.
Playing with Napier's Bones can help students strengthen their single-digit fluency while also helping them to better understand multi-digit multiplication. This is a great way to introduce students to multi-digit multiplication as they will build confidence as they easily compute the product of very large numbers. In addition, students will begin to build an intuitive understanding of place value that will transfer directly to partial products or the standard algorithm.
John Napier. MacTutor Archives. Available at https://mathshistory.st-andrews.ac.uk/Biographies/Napier/.
Create Your Own Bones!
Multiplication with Napier's Bones
Division with Napier's Bones
Let's Play: Our Digital Bones
Conclusion
Napier's Bones is a great tool for improving student fluency in multiplication as well as introducing multi-digit multiplication to students.
In creating your own set of bones, students are practicing their skip counting and identification of multiples. As they play with their bones they are gaining more and more exposure to the common single-digit products.
Napier's Bones also supports students in better understanding place value. The joints created help make sense of carrying and the ability to multiply really big numbers very quickly is a huge confidence booster before formally introducing 2-digit multiplication. The bones focus on structure. As students develop a deeper understanding of the underlying structure of multiplication, they will gain fluency and more familiarity. These activities also support exploration and discovery. Remind students of Table Mash Up from MathBait™ Multiplication Part 3. Napier did exactly what students did, explored different ways of viewing the multiplication table and an amazing invention was the result. As you continue to move forward with multi-digit multiplication, lean on the information in MathBait™ Multiplication Part 4. Encourage students to try to develop their own methods or tools for multiplying.
Next week, we are providing loads of fun activities using Napier's Bones and games that support further exploration of multiplication rods. We hope you'll join us as we dive into our penultimate unit, MathBait™ Multiplication Part 6.
Don't forget to pick up your copy of our award-winning novel to tackle middle school math through storytelling and enjoyment!
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