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Mathematics Then and Now

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MathematicsThen and Now

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°)

- clocks (60 seconds, 60 minutes or

Tablet with numbers

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

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 22

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.

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

150

Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 53.

2100

4200

Stop when the number of the left (16) is more than half of the other number you are multiplying (18).

8400

16800

Look for numbers on the left that add up to 18 (2 and 16).

900

Cross out the other rows of numbers.

Add up the remaining numbers on the right to get the final answer.

Write 1 and 76, meaning 1 lot of 76.

176

Work down, doubling the numbers, so that you’ve now got 2, 4, 8, 16, etc. lots of 76.

2152

4304

Stop when the number of the left (32) is more than half of the other number you are multiplying (39).

8608

161216

Look for numbers on the left that add up to 39 (1, 2, 4 and 32).

322432

Cross out the other rows of numbers.

2964

Add up the remaining numbers on the right to get the final answer.

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?

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

43

+ 25

+ 8

Add the partial sums

(60 + 8)

Partial Sums

Add the tens (40 + 20)

60

Add the ones (3 + 5)

68

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

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

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

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?

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)

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

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.

+

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

Multiply 4 X 20 (ones by tens)

28

Multiply 7 X 4 (ones by ones)

Add the results

1,728

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

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

2

52

47

63

28

+ 16

10

10

20

10

10

26

6

20

6

We can multiply by four simply by doubling the value twice:

37 x 4

115 x 4

double

double

74

230

double again

double again

148

460

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

take half

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

1+4

4+2

8

5

5

2

35 x 11 =

3

142 x 11 =

1

6

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.