Napiers Bones For remote learning Teacher guide Task
Napier’s Bones For remote learning
Teacher guide: Task overview This task seeks to develop students’ appreciation of the historical imperative for calculation. The task will provide students with alternative approaches to calculation that will allow them to verify their answers in a different way. Ideally students will make an informed choice of calculation strategy from a range of appropriate alternatives. Not all strategies will be equally appropriate in all circumstances.
Teacher guide: Remote learning challenges and strategies § In this task students are guided through the process of creating a set of Napier’s Bones and shown how they are used to perform calculations. § The task uses an alternative approach to consolidate students’ understanding of calculation which is scaffolded using concrete materials. § Students are encouraged to demonstrate their understanding of the properties of multiplication properties in writing. § The task focuses on developing fluency, understanding and reasoning. § Strategies for remote learning (see table). Asynchronous delivery Students collaborate and communicate with each other Buddy students up to work together via email or across communication platforms Use chat features in webex for students to work together Use online platforms for students to collaborate on journaling – eg. Google docs, One. Note Students feedback their understanding to the teacher Use online tools such as google forms – students submit their answers to prompt questions in their own time Use of school LMS systems – submissions of ideas Use the google docs or one note strategy mentioned above – teacher will be able to see what students are doing in real time and provide feedback Students document their work If using offline hard copy journals, students can take photos and send these to the teacher Students submit their work for teacher feedback Students submit their completed journal through either email or communication platform.
Links to the Victorian Curriculum strand (s) Number and algebra Curriculum substrand(s) Number and place value Levels addressed 5 -6 § This task is aimed at levels 5– 6 § What are the key learning outcomes for this task? § Students will develop an appreciation of the historical importance of calculation. § Students will create a set of Napier’s Bones. § Students will expand their repertoire of methods for performing calculations. § Students will communicate their understanding in written form (with support). § What would be some common misconceptions or difficulties that teachers need to keep an eye out for? § It may be necessary to explain it is not intended for Napier’s approach to replace other methods. § Students who rely on a rote-learned algorithm may be uncomfortable deviating from the algorithm. This task may help consolidate understanding. § Developing estimation skills provides a mental check for results obtained using a calculator.
Explore Solutions The full set of Napier’s Bones is shown opposite. • What does the number in the upper triangle represent? This is the ‘tens’ value of the product being calculated • What does the number in the lower triangle represent? This is the ‘ones’ value of the product being calculated For example: 5 x 5 = 2/5 and 5 x 9 = 4/5
Going Further Solutions You might also like these Maths 300 lessons … 103. Palindromes 052. Multo 178. Tables for 25 The idea behind Napier’s Bones (or lattice multiplication) can be readily extended. Consider: • How might you perform a multi-digit multiplication? Partition the number by place value and then calculate ones, tens etc. separately (just as in standard algorithm). • How might you perform calculations involving decimals? Multiply by an appropriate power of 10 to remove decimals. Replace the decimal point in the answer. • Perform each of the following multiplications: Examples 3094 15330 44032 35076
Going Further Explain why there are more numbers on the index strip for large numbers (such as 9) than small numbers (such as 3). For 9 x 3 = 27, while 9 x 9 = 81. What happens to the shape of the grey triangles as you move across the rods from 0 to 9? Try to explain why this occurs. The carry digit increases as the multiplicand increases. Look at the shape of the grey triangles as you move down the 0 rod. Try to explain why this occurs. The triangle points to 0 since 0 x anything = 0. Extending prompts 09 18 27 36 45 54 64 72 81 09 18 27
Summarise Solutions § How are Napier’s Bones similar to a set of multiplication tables? Each ‘bone’ contains the products expressed separately as tens (upper triangle) and units (lower triangle) digits. § Write a written explanation of how you can use Napier’s Bones to calculate 7 x 43. Place the 4 and 3 next to the index rod as shown. § How would you use Napier’s Bones to do a multi-digit calculation, such as 17 x 43? Since 17 = 10 + 7, this is calculated as: (10 x 43) + (7 x 43) = 430 + 301 = 731 § How is this similar to the written algorithm you would use for long multiplication? It really is no different; both utilise the distributive property in the same way.
Napier’s Bones The problem: In the late 1500 s people needed to find a way to make doing mathematical calculations easier. The answer was a discovery by Scottish mathematician John Napier. Learning intentions § You will learn how to perform calculations using a set of Napier’s Bones. § You will expand your repertoire of methods for performing calculations. § You will write an explanation of how to perform a calculation using Napier’s Bones. Success criteria § You will provide a clear explanation of how Napier’s Bones can be used to perform a calculation.
Getting Started John Napier (1550– 1617) was a Scottish mathematician who discovered an ingenious method of performing calculations known as logarithms, which are still used today. But Napier also discovered another way of calculating which became known as Napier’s Bones (or lattice multiplication). A set of Napier’s Bones consists of 10 rods numbered 0 to 9. Each rod has 8 squares each of which is divided into an upper and lower triangle. What do you notice about the numbers that appear on the ‘ 5’ rod and the ‘ 9’ rod? There is also an index rod that is placed on the left of the calculation.
Getting Started Enabling prompt 1 The numbers on each of Napier’s Bones are related to the number on the top of each rod. Fill in missing spaces in the table. Some of the answers have already been filled in for you. The numbers that should be placed in the green squares already appear somewhere on the chart. Can you find them?
Getting Started Enabling prompt 2 The numbers on each of Napier’s Bones are related to the number on the top of each rod. Fill in missing spaces in the table. Some of the answers have already been filled in for you.
Explore The task in detail ▶ Fill in the numbers to create a full set of Napier’s Bones. The ‘ 5’ and ‘ 9’ rods have been done to get you started. • What does the number in the upper triangle represent? • What does the number in the lower triangle represent? ▶ Once you have filled in the table, print it out and cut it into strips vertically to create an index strip and a set of ‘bones’ numbered 1 to 9. Positioning the ‘ 5’ and ‘ 9’ bones as shown allows you to multiply by 59. This is how 5 x 59 is calculated: 2 9 5
Further Examples 1. Multiplying by 59 Positioning the ‘ 5’ and ‘ 9’ bones as shown allows you to multiply by 59. This is how we calculate 6 x 59: 3 5 4
Further Examples 2. Multiplying by 95 Swapping the ‘ 5’ and ‘ 9’ bones allows you to multiply by 95. Complete the calculation for 6 x 95:
Further Examples 3. Multiplying by 599 3 -digit multiplications are achieved in a similar way. Here we see that 2 x 599 is: 1 1 9 8
Going Further Extending prompts The idea behind Napier’s Bones (or lattice multiplication) can be readily extended. Consider: • How might you perform a multi-digit multiplication? You might also like these Maths 300 lessons … 103. Palindromes 052. Multo 178. Tables for 25 • How might you perform calculations involving decimals? • Perform each of the following multiplications: Examples
Going Further Example Calculate 6 x 427 Extending prompts You might also like these Maths 300 lessons … 103. Palindromes 052. Multo 178. Tables for 25 The answer is read from right to left, following the direction indicated by the grey triangle. In this case, 6 x 427 = 2562.
Going Further Explain why there are more numbers on the index strip for large numbers (such as 9) than small numbers (such as 3). What happens to the shape of the grey triangles as you move across the rods from 0 to 9? Try to explain why this occurs. Look at the shape of the grey triangles as you move down the 0 rod. Try to explain why this occurs. Extending prompts
Summarise Record your ideas ▶ How are Napier’s Bones similar to a set of multiplication tables? ▶ Write a written explanation of how you can use Napier’s Bones to calculate 7 x 43. ▶ How would you use Napier’s Bones to do a multi-digit calculation, such as 17 x 43? ▶ How is this similar to the written algorithm you would use for long multiplication? Write your responses using the strategy outlined by your teacher.
Reflections ▶ What did you enjoy about this task? ▶ What problem solving strategies did you use? ▶ What have you learned that will help you to solve other problems in the future? Complete your final report for your teacher and submit it using the strategy outlined.
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