Numerations Systems

999nnn IIIIII

9n n IIIII125 was written as

336 would be written as

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 =

Base 10

• Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks

• Homework Pg 159 # 1,2,13,14,16,17,18,26-29

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

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 =

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 =

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 =

Convert to Base 10 notation

413 five 2004 five

100 five 214 seven

213 six 415 eight

2E4twelve 1TEtwelve

Homework

Pg 165 # 1,5,9,10,11,13

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

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

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

Use Mats, strips, and Units

• 457 + 265 =

Subtraction Algorithm

• 9 – 5 =• 9 take away 5 =

3 mats, 7 strips, 5 units -2 mats, 4 strips, 3 units

1 mat, 3 strips, 2 units

375 – 189

• • =

375 – 18936 (15) – 189

• •

375 – 18936 (15) – 189

• • =

375 – 18936 (15) – 189

2(100) + 16(10) + 15(1)

• •

375 – 189

2(100) + 16(10) + 15(1) – 189=186

• •

•

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

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

Adding & Subtracting in Other Bases

143 five

+234 five

25s 5s 1s 1 4 3 five

+ 2 3 4 five

3 12 12 five

4 3 2 five

Add

231 five 434 five

144 five 243 five

430 five 1232 five

Subtract

231 five 434 five

144 five 243 five

32 five 141 five

What Base ?

231 344 534 +414 +143 +245 1200 1042 1112

1020 572 - 203 - 144 312 427

HomeworkPg 181 #1a,3,5,6,7,21,33-36,

Multiplying Whole Numbers

9 times 3 = 3+3+3+3+3+3+3+3+3

2 groups of 10 and 7 ones = 27

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

Multiplying Whole Numbers

3 times 213 = 213+213+213• Use Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks

Multiplying Whole Numbers

4 times 243 = 243+243+243• Use Sticks in Bundles• Unifix Cubes• Units, Strips, Mats• Base Ten Blocks

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

Final Algorithm

1 2

324x 15

1620 324

4860

x 300 20 4

10 3000 200 40

5 1500 100 20

Sum = 4860

324 x 15

3 2 4

1

5

463 x 34

4 6 3

3

4

527 x 43

5 2 7

4

3

DivisionA ÷ B A/B B A

• Meaning of Division:– How many groups?– How many in each group?

How many groups?

• A ÷ B• 8 ÷ 2• 8 objects are divided into groups with 2

objects in each group.

• “How many Groups?”

How many in each group?

• A ÷ B• 8 ÷ 2• How many objects are in each group when 8

objects are divided into 2 groups.

Division Algorithms

• Standard Method• Scaffold Method• Repeated Subtraction

Homework

• Pg 192 # 1a,7,10,11,13,14,17,35-38• Pg 183 #21,22,23a,d,e,• Finish Wkst.

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

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.

Homework

• Pg 206 # 1,2,4,5,28 – 31• Review Packet on Web