Napier's bones

This activity uses an interesting historical context to help students to understand place value ideas as they explore multiplication.

For question 2, the students will find that information about John Napier is readily available in reference books and on the Internet. He lived from 1 550 to 1 617.

Napier’s invention made multiplication much easier for the people of his time. Patterns of numbers were carved on rods or sticks, but originally the facts were carved on bones and therefore became known as Napier’s bones. Napier has also been credited with the invention of the decimal point and tables of logarithms.

Ask the students to investigate what else John Napier invented.

When the students are looking for patterns to answer question 1, encourage them to look at the number at the top and scan down each strip.

• What do they notice?
• How are the two-digit numbers written? (They are split, with the tens digit above the diagonal and the ones digit below the line.)

Encourage the students to look at the following:

• the vertical columns, rows, and diagonals
• just the ones place and then just the tens place, noting their position in relation to the diagonal lines
• a symmetrical pattern of numbers or repeating patterns of numbers, and so on.

In question 3, the students use the strips to solve simple multiplication problems. When they investigated the patterns in the strips (question 1), they probably noticed that the multiples of a number go down the strip with that number at the top. For example, reading vertically down the strip with 3 at the top, you have the multiples of 3, that is, 3, 6, 9, 12, 15, and so on. So to find 4 x 3, they go down to the fourth square on the 3 strip and they get the answer, 12.

Instead of counting down the strip, the students could use the 1–9 strip as a quick way of finding the other factor. For example, to find 7 x 3, using the 3 strip:

As an extension after the students have finished the problems on the page, they could use the strips to find the value of larger multiplication problems. For example, for 78 x 46:

Note: Calculate 78 x 40 as 78 x 4 on the strip and then put a zero in the ones place of your answer and move all the digits one place to the left.

• 78 x 46 = 468 + 3 120 = 3 588

The students could then try others, such as:

• 48 x 93 = ?
• 80 x 32 = ? 
• 69 x 96 = ?
• 420 x 33 = ? 
• 1 563 x 47 = ?

1.

Answers will vary. You should see patterns if you look at the following:

• the number at the top and scan down each strip or if you look across the strips
• how the two-digit numbers are written
• symmetrical patterns
• repeating patterns.

2.

Research activity. The facts that make up Napier’s invention were originally carved on bones.

3.

Answers will vary according to the multiplication statements used. The answers to each times table up to 9 are found vertically and horizontally. For example, look at the 5 column. Notice that each entry as you look down the column is a multiple of 5 (5, 10, 15, and so on). To find 6 x 5, the sixth entry is 30, which is 6 x 5. Likewise, to find 8 x 5, look for the eighth entry in the column.

To find 8 x 7, look at the eighth entry in the 7 column. Each of the columns works in this way.

4.

Each line of the working has the same numbers in it as the arrowed strips, that is, 3 tens and 5 ones and 1 hundred and 5 tens. You are also combining 7 x 5 on the seven strip (35) with 5 x 30 on the three strip (150).

5.