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Understanding Base Systems: Why 35 in Base 7 is Greater than 35 in Base 6

In this week's agenda, we'll delve into the concept of numeral systems by exploring why the value of 35 in base 7 exceeds that in base 6, without performing any calculations. We'll discuss multiplication models through repeated addition, area, measured units, Cartesian products, and arrays. Additionally, students will engage in critical thinking as they analyze examples such as Egyptian Duplation and Lattice Multiplication. The assigned homework will include exercises focused on understanding multiplication algorithms, due Wednesday and Friday.

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Understanding Base Systems: Why 35 in Base 7 is Greater than 35 in Base 6

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  1. *sign in! (are you here?) Week 6 Warm up Problem: Without doing any calculation, explain why 35 base 7 > 35 base 6.

  2. Agenda • Discuss Exams • Multiplication Models • Begin Exploration 3.13 • Assign Homework

  3. 4 feet 2 feet Meanings for Multiplication • Repeated addition: • 3 • 4 = 4 + 4 + 4 = 12 • 4 • 3 = 3 + 3 + 3 + 3 = 12 • Area: 4 • 2 =

  4. Shirt 1 Shirt 2 Shirt 3 J B R K C J B R K C J B R K C Meanings for Multiplication • Measured Units: 5 times x units long • Cartesian Product: combinations e.g., 3 shirts, 5 pair pants

  5. Meanings for Multiplication • Arrays: 3 • 5

  6. Meanings for Multiplication • Rate: 8 miles per hour for 5 hours: • 8 • 5 = 40 miles

  7. Exploration 3.13 • First, read through the Egyptian Duplation example. Focus on the Hindu-Arabic numerals. (You can ignore the Egyptian Numerals.) • With a partner, can you explain what is going on here?If so, can you explain why it works? • With your partner, see if you can do 14 • 41; 65 • 17; 18 • 37. Do not use a calculator!!

  8. Homework • Due Wednesday • Read Section 3.3 (up to page 178 – the multiplication algorithm) • Class Notes Packet 1. Do Children’s Thinking Activity 1 (pp. 1-3). Write your answers directly in the packet. • Due Friday • Exploration 3.13 1. Do Egyptian Duplation and Lattice Multiplication for the following problems:(a) 19 • 73 (b) 26 • 50 (c) 7 • 94 2. Try to write in words why mathematically each algorithm works.

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