Calculating Bones
When John Napier, a famous mathematician, invented his “Bones” or “Rods” in the early 17th century, modern calculating was born. With this simplified recreation, students can multiply large numbers quickly, with little effort and no electricity!
Standards
Idea Sheets are cross-referenced to subjects listed in the Common Core, Next Generation Science Standards, and California Content Standards.
Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 3||Mathematical Practices|||3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 ? 4 = 24 is known, then 4 ? 6 = 24 is also known. (Commutative property of multiplication.) 3 ? 5 ? 2 can be found by 3 ? 5 = 15, then 15 ? 2 = 30, or by 5 ? 2 = 10, then 3 ? 10 = 30. (Associative property of multiplication.) Knowing that 8 ? 5 = 40 and 8 ? 2 = 16, one can find 8 ? 7 as 8 ? (5 + 2) = (8 ? 5) + (8 ? 2) = 40 + 16 = 56. (Distributive property.)||Common Core Mathematics||Grade 3||Operations And Algebraic Thinking||Understand Properties Of Multiplication And The Relationship Between Multiplication And Division|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 4||Mathematical Practices|||4.NBT.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.||Common Core Mathematics||Grade 4||Number And Operations In Base Ten2||Use Place Value Understanding And Properties Of Operations To Perform Multi-Digit Arithmetic|||4.OA.4. Find all factor pairs for a whole number in the range 1?100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1?100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1?100 is prime or composite.||Common Core Mathematics||Grade 4||Operations And Algebraic Thinking||Gain Familiarity With Factors And Multiples|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 5||Mathematical Practices|||5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.||Common Core Mathematics||Grade 5||Number And Operations In Base Ten||Perform Operations With Multi-Digit Whole Numbers And With Decimals To Hundredths|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 6||Mathematical Practices|||6.NS.2. Fluently divide multi-digit numbers using the standard algorithm.||Common Core Mathematics||Grade 6||The Number System||Compute Fluently With Multi-Digit Numbers And Find Common Factors And Multiples|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 7||Mathematical Practices|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||Grade 8||Mathematical Practices|||Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. ||Common Core Mathematics||High School||Mathematical Practices
2.4 Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = __).||CA Mathematics||Grade 3||01. Number Sense||2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division.
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