numerations systems 999nnn iiiiii 9n n iiiii 125 was written as 336 would be written as
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Numerations Systems
999nnn IIIIII
9n n IIIII125 was written as
336 would be written as
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Roman Numerals
I V X L C D M1 5 10 50 100 500 1000
M D CCC L V I
IV = 4 IX = 9 XL = 40 CD = 400 CM=
M CM L IX =
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Base 10
• Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks
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• Homework Pg 159 # 1,2,13,14,16,17,18,26-29
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Different Base Systems
Base FiveDigits used 0,1,2,3,4Decimal Value of Positions54 = 625 53 = 125 52 = 25 51 = 5 50 = 1 1 3 2 4 2Five
625 + 3 x 125 + 2 x 25 + 4 x 5 + 2 x 1 = 625 + 375 + 50 + 20 + 2 = 1072
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Base 6
Base 6Digits used 0,1,2,3,4,5Decimal Value of Position64 = 1296 63 = 216 62 = 36 61 = 6 60 = 1 1 2 3 5 2six =
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Base Seven
Base 7Digits used 0,1,2,3,4,5,6Decimal Value of Position74 = 2401 73 = 343 72 = 49 71 = 7 70 = 1 1 2 3 5 2seven =
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Base 2
Base 2Digits used 0,1Decimal Value of Position25 = 32 24 = 16 23 = 8 22 = 4 21 = 2 20 = 1 1 0 1 1 0 1two =
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Convert to Base 10 notation
413 five 2004 five
100 five 214 seven
213 six 415 eight
2E4twelve 1TEtwelve
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Homework
Pg 165 # 1,5,9,10,11,13
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Addition Algorithm
• Making exchanges with Units, Strips, Mats• 15 units, 11 strips, and 2 mats. What
exchanges must be made in order to represent the same number but with the smallest number of pieces?
• Pg 166
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Units, Strips, Mats
Adding
1 mats, 3 strips, and 5 units+ 2 mats, 4 strips, and 3 units
3 mats, 7 strips, and 8 units
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Add
3 mats, 5 strips, 7 units+ 2 mats, 7 strips, 4 units
5 mats, 12 strips, 11 unitsExchange5 mats, 13 strips, 1 unitExchange6 mats, 3 strips, 1 unit
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Use Mats, strips, and Units
• 457 + 265 =
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Subtraction Algorithm
• 9 – 5 =• 9 take away 5 =
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3 mats, 7 strips, 5 units -2 mats, 4 strips, 3 units
1 mat, 3 strips, 2 units
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375 – 189
• • =
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375 – 18936 (15) – 189
• •
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375 – 18936 (15) – 189
• • =
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375 – 18936 (15) – 189
2(100) + 16(10) + 15(1)
• •
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375 – 189
2(100) + 16(10) + 15(1) – 189=186
• •
•
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324- 156
100s 10s 1s 3 2 4- 1 5 6
3 1 14- 1 5 6
2 11 14 - 1 5 6
1 6 8
Exchange
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3 hours 24 minutes 54 seconds + 2 hours 47 minutes 38 seconds
6 hours 12 minutes 32 seconds
7 hours 46 minutes 29 seconds - 3 hours 27 minutes 52 seconds
4 hours 18 minutes 37 seconds
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Adding & Subtracting in Other Bases
143 five
+234 five
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25s 5s 1s 1 4 3 five
+ 2 3 4 five
3 12 12 five
4 3 2 five
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Add
231 five 434 five
144 five 243 five
430 five 1232 five
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Subtract
231 five 434 five
144 five 243 five
32 five 141 five
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What Base ?
231 344 534 +414 +143 +245 1200 1042 1112
1020 572 - 203 - 144 312 427
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HomeworkPg 181 #1a,3,5,6,7,21,33-36,
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Multiplying Whole Numbers
9 times 3 = 3+3+3+3+3+3+3+3+3
2 groups of 10 and 7 ones = 27
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Multiplying Whole Numbers
9 times 3 = 3+3+3+3+3+3+3+3+3• Use Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks
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Multiplying Whole Numbers
3 times 213 = 213+213+213• Use Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks
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Multiplying Whole Numbers
4 times 243 = 243+243+243• Use Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks
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15 x 324
324 15 20
100 1500 40 200 3000 4860
5 x 4
5 x 20
5 x 300
10 x 20
10 x 4
10 x 300
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Final Algorithm
1 2
324x 15
1620 324
4860
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x 300 20 4
10 3000 200 40
5 1500 100 20
Sum = 4860
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324 x 15
3 2 4
1
5
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463 x 34
4 6 3
3
4
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527 x 43
5 2 7
4
3
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DivisionA ÷ B A/B B A
• Meaning of Division:– How many groups?– How many in each group?
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How many groups?
• A ÷ B• 8 ÷ 2• 8 objects are divided into groups with 2
objects in each group.
• “How many Groups?”
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How many in each group?
• A ÷ B• 8 ÷ 2• How many objects are in each group when 8
objects are divided into 2 groups.
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Division Algorithms
• Standard Method• Scaffold Method• Repeated Subtraction
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Homework
• Pg 192 # 1a,7,10,11,13,14,17,35-38• Pg 183 #21,22,23a,d,e,• Finish Wkst.
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Mental Arithmetic & Properties
• 35 + 7+ 15 (Comm.)• 8+3+4+6+7+12+4+3+6+3 (add to 10s)• 25 x 8• 4 x 99
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Rounding Numbers
• Determine to which position your are rounding
• If the digit immediately to the right of this position is 5 or more, add 1 to the digit in the position to which you are rounding. Otherwise leave the digit unchanged.
• Replace with 0s all digits to the right of the position to which you are rounding, up to the decimal point.
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Homework
• Pg 206 # 1,2,4,5,28 – 31• Review Packet on Web