MathBait™ Multiplication Part 5: Multiplication with Napier's Bones
- MathBait
- Apr 1, 2024
- 2 min read
Updated: Apr 14, 2024
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!
You can purchase a set of well-crafted Napier's Bones (Our set shown above is from Creative Crafthouse), or you can play with our digital version (below). However, the process of making your own set of Napier's Bones is fun, a great way to introduce rod multiplication, and helps students continue to build fluency with multiples.
In a time crunch? Scroll down to find our printable version. A fun (but more flimsy) way to quickly create a set of bones.
Materials
Popsicle Sticks (~30 per set of bones)
Markers
Ruler
Paper
Step 1: Measure each Popsicle Stick
Once you have collected the Popsicle sticks you will use, start by measuring a stick. This will help to ensure your rods are properly aligned. While the box may provide the measurements, we do not include the rounded edges in our calculations. This helped to avoid a wonky final value.
We used standard sticks from Creatology and measured them to be about 13 cm (excluding the rounded edges).
Once you have your measurement, divide it by 10. This will give you the length of each segment. We recommend making about 20 bones per set, two for each digit.
Step 2: Mark Your First Stick
As our stick, not including the rounded ends, measured to be about 13 cm, each segment will be 1.3 cm in height.
Take one stick (this will be your guide stick) and mark off 10 lines based on your measurement. In the image above, we have marked 10 lines each creating a cell of height 1.3 cm.
If working in a larger group or with young students who may have difficulties measuring, we recommend making a few guiding sticks to distribute around the room. Students will be able to complete the remaining steps independently or with minor assistance.
Step 3: Partition Your Sticks
Line up several Popsicle sticks and place the guide stick on the left. Place your ruler flush with the first mark on the guide stick. Finally, use a marker to draw a straight line across each stick.
Repeat the process for each of the marks on your guide stick.
Step 4: Divide Each Cell Diagonally
Now that your sticks are partitioned, we'll draw a diagonal line through each cell. Line up your ruler to create a diagonal from the top left to the bottom right of each cell.
Step 5: Add in the Numbers
Start by adding a digit to the top of each stick (0, 1, 2, 3, ..., 9). We recommend making two of each for a total of 20 sticks.
Next, students will add in the multiples of the digit at the top of the stick. This is a great chance for students to continue to build fluency by working on their skip counting.
In order to create the perfect joints, one multiple will be placed in each cell. The value in the ten's place will go above the diagonal while the value in the one's place will go below.
For instance, if making the 3-rod, the first entry will be 0/3 as 3×1=3. We place a 3 in the ones digit, below the diagonal line, and a 0 in the tens digit, above the diagonal line. The remaining entries on this stick will be 0/6, 0/9, 1/2, 1/5, 1/8, 2/1, 2/4, 2/7.
Repeat for each stick or "bone". Plain Popsicle sticks and darker colored markers work best for maximum contrast. Colored sticks can be helpful to organize information (for example: 2's are on yellow sticks, 3's on blue, etc.), however they require a very dark marker for contrast and may be more difficult to read.
Step 6: Create Your Lead Bone
The final step is to create a lead bone. This is a single Popsicle stick with the digits 1-9 written to align with each cell.
Congratulations! You've made your very own set of Napier's Bones!
Don't have Popsicle sticks? Not to worry! We've included a printable version below with both empty and completed rods.
Don't forget to check out our digital bones! This allows students to play anywhere and offers step-by-step support and an addition helper for carrying and working with larger values.
Multiplication with Napier's Bones
Now that you have your own set of bones, it's time to play with them and create some serious math magic. You might be surprised with how quickly and easily you can multiply large values. Grab your bones or open our digital game in a new window to practice along with the steps.
Step 1: Collect Your Bones
The first step in multiplying is to collect the bones you need. You'll find it's easiest to collect the bones for the larger value. For instance, if multiplying 1967×8, we'll collect the 1, 6, 7, and 9 bones.
Next, line up your bones up to create the number you wish to multiply. Since we are multiplying 1967 we will arrange our bones in this order.
Step 2: The Lead Bone
Place your lead bone to the left and line it up with the bones you have collected. We need to determine what row to focus on. If multiplying a multi-digit number, start with the largest place value. For instance, if multiplying 1967×58, we would focus on the 5th row first. Since we are multiplying 1967×8, we'll focus on the 8th row.
Set 3: Sum the Joints
The design of the bones create "joints". The first and last are always triangles, while the middle joints form parallelograms. For 1967×8, our first joint is a 6, so we transcribe this value. The next joint shows us an 8 and a 5. Since 8+5=13, we transcribe a 3 and carry a 1 over to the next joint.
Next, we have 2 and 4. Adding the values gives us 6, plus the 1 from the previous joint gives us 7. So far we have 763 transcribed. The penultimate joint shows us 8+7=15, again we transcribe the 5 and carry the 1 over to the final joint and end up with 15736. This is the value of 1967×8.
We selected a more complex example to cover all cases, however, there are many multiplication problems students can complete by simply looking at a set of bones. For instance, 7512×6 can be found to be 45,072 with just a glance! You got this.
To practice, try out our digital bones. This highlights each joint (as in the images above) and provides carrying and addition support. This is a great way to help students become familiar with using the bones.
Multiplying Larger Values
What about 1967×58, or even larger values? Place value plays a huge role in multiplying larger values. Just as we did above, we would reference the 8th row, but we would also need to reference and transcribe the value from the 5th row.
Noting that multiplying by 58 is really multiplying by 50+8, we must place our transcribed value for the 5th row into the ten's place. To find the final product we then simply add our two transcribed values. Let's look at an example with 436×82.
Unlike the standard algorithm, Napier begins with the largest digit. After we collect our 4, 3, and 6 bones and align them with our lead bone, we will first focus on the 8th row.
This has no carrying. From simply looking at the bones we see that 436×8=3488. As our goal was to multiply 436×82, we must note that we were interested in multiplying 436×80 rather than 436×8. Thus, we must transcribe 3488×10=34880.
Next, we shift our focus to the second row in order to multiply 436×2.
Again, we can see the value quite easily, 436×2=872.
To complete our product, we must sum both values. Notice how this process is simply applying the Distributive Property that students were introduced to in MathBait™ Multiplication Part 3! This exercise is great practice for both addition of large numbers and using the distributive property.
So what is 436×82? Summing our transcribed values of 34880 and 872 we find
436×82=35752.
The larger the factors, the more values to sum. Thus, Napier's Bones can require some pen and pencil work. Our digital bones includes an addition helper to keep track of place value and carrying allowing young students to find products with ease.
Division with Napier's Bones
We can divide with Napier's Bones too! We recommend waiting to formally introduce dividing to students until they are very fluent in multiplication. However, a great introduction is using activities like Emoji Mystery and Missing Numbers from MathBait™ Multiplication Part 3. This allows students to problem-solve through questions such as: if #×8=48, what is #? This provides an inroad to explain finding a missing factor in a product is the same as asking what is 48÷8?
Step 1: Collect your Bones
In division, we collect the bones for the number we are dividing by. For instance, if dividing 4672÷36, we'll collect the bones 3 and 6.
Step 2: Examine the Multiples
By placing the bones 3 and 6, we can quickly scan down to view all the multiples of 36.
The bones to the right show us the multiples of 36 are 36, 72, 108, 144, 180, .... Since our goal is to divide 4672, we look for the first multiple less than 46. As this is 36, we transcribe a 1, and place the multiple (36) under 46, adding any trailing zeros.
Step 3: Subtract
The final step is to subtract. We have 4672-3600=1072.
The value 1072 is our remainder. We repeat steps 2 and 3 until the remainder is less than the value we are diving by.
Steps 4 and on: Repeat
Using our remainder of 1072, we now look to the bones for the largest multiple of 107. Finding 72 (36×2), we transcribe a 2 and subtract.
Note in Step 1 we transcribed a 1, so we currently have "12" as our tally.
Subtracting 1072-720=352, we again return to the bones to find the largest multiple of 36 less than 352. This leads us to row 9 as 36×9=324. We transcribe the 9 giving us 129.
In our final subtraction we find 352-324=28. As 28 is less than 36, this is our final remainder. We have found 4672÷36=129 with a remainder of 28 or 129 and 28/36 which simplifies to 129 and 7/9.
Napier's Bones with division takes some practice. However, it is a great activity to simply play with the bones, as shown here, to see how we can also find all the multiples (up to 9) of larger values such as 36 too!
Let's Play: Our Digital Bones
Ready to play with the bones? Our digital bones will take students step-by-step on how to multiply and divide. In addition, there is a "test your skills" feature for students to practice using the bones. We provide support in carrying and addition. Watch the video below to explore how to use the digital bones.
Play
Note: If playing on a tablet, we recommend tilting the device vertically for the best experience. If using the Safari mobile app, full screen mode is not available.
©MathBait™ created with GeoGebra
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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