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MathBait™ Multiplication

Divide with Napier's Bones

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Division and multiplication are really synonymous. As students advance they will learn dividing by 3 is akin to multiply by 1/3. Thus, the powerful multiplication ability of Napier's Bones can be extended into division as well! In this activity we'll show you how to harness the strength of the bones to find quotients as well!


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Napier's Bones







Help students better understand division with Napier's Bones! 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

Napier's Bones 36

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.

Steps showing division using Napier's Bones

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!

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