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Most notable advancements in the early development of mathematics: • Mayans • Babylonians • Egyptians • Greeks • Chinese 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°) Tablet with numbers 1 set of 3600 52 sets of 60 30 sets of 1 1 52 30 1 ˟ 3600 = 52 ˟ 60 = 30 ˟ 1 = 3600 3120 30 6750 Try to write: 23 41 82 121 82 = 60 + 22 121 = 2 ˟ 60 + 1 Babylonian multiplication concentrated on perfect squares (3)(4) = (3 + 4)2 – 32 – 42 = 49 – 9 – 16 = 24 = 12 2 2 2 Simple grouping system (hieroglyphics) The Egyptians used the stick for 1 heel bone for 10 scroll for 100 lotus flower for 1,000 bent finger for 10,000 burbot fish for 100,000 astonished man for 1,000,000. 3000 + 200 + 40 + 4 = 3244 What are the following values? 52 21,238 The Ancient Egyptians used a pencil and paper method for multiplication which was based on doubling and addition. Write down 1 and 50 Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 53. Stop when the number of the left (16) is more than half of the other number you are multiplying (18). Look for numbers on the left that add up to 18 (2 and 16). Cross out the other rows of numbers. Add up the remaining numbers on the right to get the final answer. 1 2 4 8 16 50 100 200 400 800 900 Write 1 and 76, meaning 1 lot of 76. Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 76. Stop when the number of the left (32) is more than half of the other number you are multiplying (39). Look for numbers on the left that add up to 39 (1, 2, 4 and 32). 1 2 4 8 16 32 76 152 304 608 1216 2432 Cross out the other rows of numbers. Add up the remaining numbers on the right to get the final answer. 2964 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? This chariot travels 23km in an hour. How far will it travel in 6 hours? We have seen that different civilizations had different methods to handle basic arithmetic This demonstrates that we can add, subtract, multiply, and divide numbers in multiple ways and still get the same answer 43 + 25 Add the tens (40 + 20) Add the ones (3 + 5) Add the partial sums (60 + 8) 60 + 8 68 Add the Add the Add the Add the hundreds (200 + 400) tens (60 +80) ones (8 + 3) partial sums (600 + 140 + 11) 268 + 483 600 140 + 11 751 1. Create a grid 2. Draw diagonals 3. 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 4. Add along each diagonal and record any regroupings in the next diagonal 7 + 4 1 1 2 1 8 8 1 6 6 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 +2 -2 90 + 34 +10 - 10 100 + 24 124 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? 1. Place the smaller number at the bottom of the hill and the larger at the top. 38 – 14 = 2. Start with 14, add to the next friendly number. (14+6=20) 3. Start with 20, add to the next friendly number. (20+10=30) 4. Start with 30, add to get 38. (30+8=38) Record the numbers added at each interval: (6+10+8=24) 1. Replace each digit to be subtracted with its nines complement, and then add 2. Delete the leading 1 3. Add 1 to the final result 75 – 38 75 + 61 136 +1 37 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. When multiplying by “Partial Products,” you must first multiply parts of these numbers, then you add all of the results to find the answer. Multiply 20 X 60 (tens by tens) Multiply 60 X 7 (tens by ones) Multiply 4 X 20 (ones by tens) Multiply 7 X 4 (ones by ones) Add the results 27 X 64 1,200 420 80 + 28 1,728 (20+7) (60+4) 1. Create a grid 2. Draw diagonals 3. Copy one digit across top of grid and the other along the right side 4. 5. 25 x 47 = 1,175 2 5 0 2 4 Multiply each digit in the 1 8 0 1 top factor by each digit in the side factor, placing the 1 3 7 tens digit in the upper half 4 5 1 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 7 5 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. We can add large set of numbers quickly by grouping values that add to ten 10 20 10 2 52 47 63 28 + 16 20 6 10 10 26 6 We can multiply by four simply by doubling the value twice: 37 x 4 double 115 x 4 double 74 double again 148 230 double again 460 We can multiply by five simply by multiplying by ten and then take half: 42 x 5 multiply by 10 multiply by 10 420 take half 73 x 5 730 take half 210 365 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 35 x 11 = 3 8 5 1+4 4+2 142 x 11 = 1 5 6 2 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.