Mathematics Then and Now

1. MathematicsThen and Now

2. Most notable advancements in the early development of mathematics: • Mayans • Babylonians • Egyptians • Greeks • Chinese Ancient Math

3. Wrote on tablets • Used two symbols for numbers • Ones • Tens • Used a base 60 place system • clocks (60 seconds, 60 minutes or 3600 seconds) • circle (360°) Babylonians

4. Tablet with numbers Babylonians

6. 1 set of 3600 52 sets of 60 30 sets of 1 1 52 30 • 1 ˟ 3600 = 3600 • 52 ˟ 60 = 3120 • 30 ˟ 1 = 30 • 6750 Babylonians

7. Try to write: 23 41 82 121 82 = 60 + 22 121 = 2 ˟ 60 + 1 Play with Babylonian Numbers

8. Babylonian multiplication concentrated on perfect squares (3)(4) = (3 + 4)2 – 32 – 42 = 49 – 9 – 16 = 24 = 12 2 2 2 Babylonian Multiplication

9. Simple grouping system (hieroglyphics) The Egyptians used the stickfor 1 heel bone for 10 scrollfor 100 lotus flower for 1,000 bent finger for 10,000 burbot fishfor 100,000 astonished man for 1,000,000. Egyptian Mathematics

10. 3000 + 200 + 40 + 4 = 3244 What are the following values? 52 21,238 Egyptian Mathematics

11. The Ancient Egyptians used a pencil and paper method for multiplication which was based on doubling and addition. Egyptian Multiplication

12. Write down 1 and 50 1 50 Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 53. 2 100 4 200 Stop when the number of the left (16) is more than half of the other number you are multiplying (18). 8 400 16 800 Look for numbers on the left that add up to 18 (2 and 16). 18 x 50 900 Cross out the other rows of numbers. Add up the remaining numbers on the right to get the final answer.

13. Write 1 and 76, meaning 1 lot of 76. 1 76 Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 76. 2 152 4 304 Stop when the number of the left (32) is more than half of the other number you are multiplying (39). 8 608 16 1216 Look for numbers on the left that add up to 39 (1, 2, 4 and 32). 39 x 76 32 2432 Cross out the other rows of numbers. 2964 Add up the remaining numbers on the right to get the final answer.

14. This jar holds 17 litres of water.How much water will 25 jars hold? A potter makes 35 pots each month. How many will he make in a year? Now it’s your turn! This chariot travels 23km in an hour. How far will it travel in 6 hours?

15. This demonstrates that we can add, subtract, multiply, and divide numbers in multiple ways and still get the same answer We have seen that different civilizations had different methods to handle basic arithmetic Arithmetic

16. 43 + 25 + 8 Add the partial sums (60 + 8) Partial Sums Add the tens (40 + 20) 60 Add the ones (3 + 5) 68

17. 268 Add the hundreds (200 + 400) + 483 + 11 Add the partial sums (600 + 140 + 11) Partial Sums 600 Add the tens (60 +80) 140 Add the ones (8 + 3) 751

18. Lattice Sums 7 8 + 4 8 Create a grid Draw diagonals Add each column, placing the tens digit in the upper half of the cell and the ones digit in the bottom half of the cell Add along each diagonal and record any regroupings in the next diagonal 1 1 1 6 1 2 6

19. The opposite change rule says that if a value is added to one of the numbers, then subtract the value from the other number 88 + 36 90 + 34 100 + 24 +10 +2 - 2 - 10 Opposite Change 124

20. Let’s look at some different methods to subtract numbers We are familiar with the basic borrowing methods, but did you know we can subtract by adding? Subtraction

21. Counting Up-Hill 38 – 14 = 24 1. Place the smaller number at the bottom of the hill and the larger at the top. 30 38 +8 2. Start with 14, add to the next friendly number. (14+6=20) +10 20 Record the numbers added at each interval: (6+10+8=24) 3. Start with 20, add to the next friendly number. (20+10=30) +6 14 4. Start with 30, add to get 38. (30+8=38)

22. 75 + 61 75 – 38 Replace each digit to be subtracted with its nines complement, and then add Delete the leading 1 Add 1 to the final result 136 +1 Nines Complement 37

23. Let’s look at some different methods to multiply numbers We have already seen two methods to multiply beyond our current procedure (Babylonian method of squares and the Egyptian method of doubles. Let’s look at a few more. Multiplication

24. + 2 7 (20+7) When multiplying by “Partial Products,” you must first multiply parts of these numbers, then you add all of the results to find the answer. X 6 4 (60+4) 1,200 Multiply 20 X 60 (tens by tens) 420 Multiply 60 X 7 (tens by ones) 80 Partial Product Multiply 4 X 20 (ones by tens) 28 Multiply 7 X 4 (ones by ones) Add the results 1,728

25. Lattice Product Create a grid Draw diagonals Copy one digit across top of grid and the other along the right side Multiply each digit in the top factor by each digit in the side factor, placing the tens digit in the upper half of the cell and the ones digit in the bottom half of the cell Add along each diagonal and record any regroupings in the next diagonal 1,175 25 x 47 = 2 5 2 0 4 8 0 1 1 3 1 7 4 5 1 7 5

26. We can often perform basic arithmetic in our head faster than we can by writing it down or plugging it into a calculator. We need to recognize certain patterns to help the process. Math Tricks

27. We can add large set of numbers quickly by grouping values that add to ten 2 52 47 63 28 + 16 10 10 20 10 10 26 Group Tens for Addition 6 20 6

28. We can multiply by four simply by doubling the value twice: 37 x 4 115 x 4 double double 74 230 Multiply by Four double again double again 148 460

29. We can multiply by five simply by multiplying by ten and then take half: 42 x 5 73 x 5 multiply by 10 multiply by 10 420 730 Multiply by Five take half take half 210 365

30. We can multiply by eleven by keeping the first and last digit and then adding digits that are next to each other to get the rest of the digits 3+5 1+4 4+2 8 5 5 2 35 x 11 = 3 142 x 11 = 1 6 Multiply by Eleven

31. Keep in mind that there is more than one way to get to the correct answer. We have shown you a few different methods to add, subtract and multiply, but there are many other methods. Try these or other methods to see if you like them. Perhaps you can invent your own. Arithmetic