Counting by Fives

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Counting by Fives
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INTRODUCTION

• These base-five pieces are called units, longs,
  flats, and blocks

Units (or ones)
long
flat
block
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INTRODUCTION

• Examine the pieces to determine the patterns
• 1 long ? units.
• 1 flat ? longs ? units.
• 1 block ? flats ? longs ? units.

ANSWER
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INTRODUCTION

• Compare your answer
• 1 long 5 units.
• 1 flat 5 longs 25 units.
• 1 block 5 flats 25 longs 125 units.

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INTRODUCTION

• If you expand the pieces, what would the next
  piece contain?

ANSWER

• 5 blocks because each base-five piece contain 5
  of the next smaller piece.

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REPRESENT NUMBERS

• To represent 7 units

with base-five pieces, we would use 1 long and
2 units
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REPRESENT NUMBERS

• To represent 36 units

with base-five pieces, we would use 1 flat, 2
longs and 1 units
This collection uses the smallest possible number
of base-five pieces.
25 5 5 1 36
units
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PRACTICE REPRESENTING NUMBERS

• Use your base-five pieces to represent each
  amount.
• Use the smallest possible number of base-five
  pieces.

(12 units)
ANSWER
2 longs and 2 units
5 5 2 12 units
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PRACTICE REPRESENTING NUMBERS
(43 units)
ANSWER
1 flat, 3 longs, 3 units
25 5 5 5 3
43 units
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PRACTICE REPRESENTING NUMBERS

• 87 units

ANSWER
3 flats, 2 longs and 2 units
25 25 25
5 5 2 87 units
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PRACTICE REPRESENTING NUMBERS

• 181 units

ANSWER
1 block, 2 flats, 1 longs and 1 units
125 25
25 5 1 181 units
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PRACTICE REPRESENTING NUMBERS

• 194 units

ANSWER
1 block, 2 flats, 3 longs and 4 units
125 25 25
5 5 5 4 194 units
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NUMERATION EXPRESSION IN BASE-FIVE

• In base-five numeration, 2 blocks, 1 flat, 3
  longs, 4 units are recorded as 2134 five
• Write each amount in base-five numeration. Use
  your previous work to help.
• 87 units
• 43 units

ANSWER
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NUMERATION EXPRESSION IN BASE-FIVE

• 87 units 3 flats, 2 longs, 2 units 322 five
• 43 units 1 flat, 3 longs, 3 units 133 five

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PRACTICE

• Model each base-five numeral with base-five
  pieces and then tell how many units are
  represented.
• Example 41five means 4 longs, 1 unit

5 5 5 5 1 21 units
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PRACTICE

• Work these problems
• 324five
• 1223five
• 1034five

ANSWER
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PRACTICE

• 324 five 3 flats, 2 longs, 4 units

25 25 25
5 5 4 89 units
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PRACTICE

• 1223 five 1 block, 2 flats, 2 longs, 3 units

125 25 25
5 5 3 188 units
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PRACTICE

• 1034 five 1 block, 0 flats, 3 longs, 4 units

125 5 5 5 4 144
units
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ADDITION

• Numbers written as base-five numerals can be
  added using base-five pieces.
• Example
• 21five 33five is represented by

Combine base-five pieces



Trade pieces to use smallest possible number of
pieces
21five 33five 104five
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ADDITION

• What does 104five mean?
• 104five means 1 flat, 0 long, 4 units.
• Does 14five mean the same thing?
• No, 14five means 1 long, 4
  units.

ANSWER
ANSWER
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PRACTICE

• Work these problems
• 112five 23five
• 421five 122five
• 231five 214five

ANSWER
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ADDITION

• 112five 23five

5 units 1 long
112five 23five 140five 1 flat, 4 longs, 0
unit

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ADDITION

• 421five 122five

5 flats 1 block
421ive 122five 1043five
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ADDITION
5 units 1 long

• 231five 214five

Continue Next Page
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ADDITION
5 longs 1 flat

• 231five 214five(cont.)

5 flats 1 block

231ive 214five 1000five
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ADDITION

• Use the patterns you found or base-five pieces to
  complete the addition table in base-five
  numeration.

ANSWER
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ADDITION

• Compare your table.

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ADDITION

• When you add using base-five numeration, what
  happens when you get 5 of one of the pieces?

You trade the 5 pieces for 1 of the next larger
unit.
ANSWER
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SUBTRACTION

• To subtract base-five numerals, use base-five
  pieces and take away the pieces that match.
• Example 24five 12five

Take away the pieces that match. The ones that
are left represent the answer.

-
24five 12five 12 five
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SUBTRACTION
We cannot take 4 away from 2 so we trade 1 long
for 5 units

• Sometimes you have to trade pieces to be able to
  take away all the pieces of the subtrahend.
• Example 32five 14five

Take away matching pieces


-
-
32five 14five 13five
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PRACTICE

• Use your base-five pieces and follow the examples
  to answer these problems
• 332five - 121five
• 213five - 132five
• 1031five - 243five

ANSWER
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PRACTICE

• 332five - 121five 211five

-

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PRACTICE

• 213five - 132five 31five

-

-

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PRACTICE

• 1031five - 243five

-

Continue Next Page
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PRACTICE

• 1031five - 243five 233five

-

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MULTIPLICATION

• We can also use base-five pieces to model
  multiplication.
• To model 2five x 4five
• Model 4five
• Double it.
• Trade pieces to use the smallest number of base
  five pieces

2five x 4five 13five
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MULTIPLICATION

• To model 3five x 142five
• Begin by modeling 142five
• Then triple it

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MULTIPLICATION

• Trade pieces to use the smallest of pieces.

3five x 142five 1031five
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MULTIPLICATION

• Use the patterns or base-five pieces to complete
  the multiplication table in base-five numeration.

ANSWER
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MULTIPLICATION

• Compare your table

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MULTIPLICATION

• Use the table to find the product
• 3five x 142five

ANSWER
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MULTIPLICATION

• Is this the same answer you got with the
  base-five pieces?

YES!
ANSWER

• Find these products. Use patterns or base-five
  pieces.
• 2five x 34five
• 3five x 123five

ANSWER
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MULTIPLICATION

• 2five x 34five

2five x 34five 123five
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MULTIPLICATION

• 3five x 123five

3five x 123five 424five
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MULTIPLICATION

• The previous problems could be worked by modeling
  with base-five piece.
• It is not practical to use base-five pieces for
  this problem
• 22five x 31five

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MULTIPLICATION

• This example uses expanded notation.
• 22five 20five 2five
• x 31five x 30five 1five
• 2
  (2five x 1five)
• 20 (20fivex 1five)
• 110 (30fivex 2five)
• 1100 (20fivex 30five)
• 1232five

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MULTIPLICATION

• Find these products.
• 13five x 12five
• 23five x 42five

ANSWER
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MULTIPLICATION

• 13five x 12five
• 13five 10five 3five
• x 12five x 10five 2five
• 11
  (3five x 2five)
• 20 (10fivex 2five)
• 30 (10fivex 3five)
• 100 (10fivex 10five)
• 211five

13ive x 12five 211five
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MULTIPLICATION

• 23five x 42five
• 23five 20five 3five
• x 42five 40five 2five
• 11
  (3five x 2five)
• 40 (20fivex
  2five)
• 220 (40fivex 3five)
• 1300 (40fivex 20five)
• 2121five

23five x 42five 2121five
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