CHAPTER
Whole Numbers
Concrete models can help students represent numbers and develop number sense; they can also
help to bring meaning to students' use of written symbol and can be useful in building placevalue concepts. But using materials, especially in a rote manner, does not ensure understanding.
Teachers should try to uncover students' thinking as they work with concrete materials by asking
questions that elicit students' thinking and reasoning. I
Activity Set
3.1
MODELS FOR NUMERATION
WITH MULTI BASE PIECES
PURPOSE
Virtual
Manipulatives
To use base-five pieces and visual diagrams for other bases to introduce the concepts of grouping
and place value in a positional numeration system.
MATERIALS
Base-Five Pieces from the Manipulative Kit or from the Virtual Manipulatives.
INTRODUCTION
www.mhhe.com/bbn
The concepts underlying positional numeration systems-grouping,
regrouping, and place
value-become
much clearer when illustrated with concrete models, and many models have been
created for this purpose. Models for numeration fall roughly into two categories or levels of
abstractness. Grouping models, such as bundles of straws and base-five blocks, show clearly how
groupings are formed; abacus-type models, such as chip-trading or place-value charts, use color
and/or position to designate different groupings.
/
Lou:
Bundles of Straws
1 Principles
Base-Five Blocks
Uilil
Place Value Mat
and Standards for School Mathematics (Re ton, VA: National Council of Teachers of Mathematics, 2000): 80.
55
Activity Set 3.1
57
Models for Numeration with Multibase Pieces
The base pieces shown below are in the Manipulative Kit and in the Virtual Manipulatives.
They are called base-five pieces because each piece, other than the unit, has 5 times as many unit
squares as the preceding piece.
o
Long-flat
Flat
Long
Unit
1. Use your base-five pieces to form the collection consisting of 8 units, 6 longs, and 5 flats, as
shown in Figure I. Exchange 5 units for 1 long,S longs for 1 flat, and 5 flats for l long-flat.
This results in the collection shown in Figure II. The two collections are equivalent because
they both contain the same number of unit squares, but the second collection has fewer basefive pieces. (The first collection has 19 pieces, and the second has 7 pieces.)
o
o
o
o
o
o
o
0
0
0
0
Figure I
Figure II
Use your base-five pieces to form each of the following collections. Make exchanges until
you have an equivalent collection with the least number of base-five piece. This is called the
minimal collection. Record the number of long-flats, flats, longs, and units in your minimal
collection in the table below.
a.
b.
I!Hllmmm
m
~ ~ ~ ~
c.
b.
*c.
0 0
0 0
0 0
0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Long-Flats
a.
0
0
0
0
UU
0 0
0 0
0
0 0
Flats
Longs
Units
58
Chapter 3
Whole Numbers
d. Explain why you never have to write a numeral greater than 4 in the table for a minimal collection of base-five pieces.
2. U ing the base-five pieces, you can represent a collection of 28 units by a minimal collection of 1 flat, 0 longs, and 3 units. This is recorded in the table below. Using base-five
pieces to aid visualization, supply the missing numbers in the following table for minimal
collections.
No. of Unit
Squares
Long-Flats
Flats
Longs
Units
a.
28
0
1
o
3
b.
31
*c.
126
d.
200
2
3
e.
0
*f.
3
3
o
3
g.
4
4
4
4
3. Minimal collections of base-five pieces can be recorded without using tables. For example, the
first entry in the table above can be written as 103five' In doing this, we must agree that the positions ofthe digits from right to left represent the numbers of units, longs, flats, and long-flats.
This method of writing numbers is called positional numeration, and 103five is called a basejive numeral. Write the base-five numerals for each of the other entries in the table in activity 2.
a. I03five
b.
*c.
e.
*f.
g.
d.
4. The base-five pieces that represent the numeral 2034five are shown here. There is a total of
269 unit squares in these pieces.
o
o
o
o
2034five
Represent the following numbers with your base-five pieces, and determine the total
number of unit squares in each.
Base-Five Numeral
a.
2304five
*b.
1032five
C.
2004five
Total Number of
Unit Squares
Activity Set 3.1
59
Models for Numeration with Multibase Pieces
5. Here are the first three pieces for base seven. Draw the first three pieces for base three and
base ten in the space provided below.
Base seven
Flat
Long
Unit
II!!
*a. Base three
b. Base ten
6. Draw a diagram of the collection of base pieces representing each of the following numbers.
Then determine the total number of unit squares in each collection.
a. 122three
*b.
42\even
c. lS7teo
Total number of unit squares
_
Total number of unit squares
_
Total number of unit squares
_
7. For each of the bases in parts a, b, and c, make a sketch of the minimal collection of base
pieces that represents the collection of unit squares shown. Then write the numeral for the
given base.
a. Base five
ODD
ODD
ODD
ODD
ODD ODD
ODD ODD
ODD ODD
ODD ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
Minimal Collection Sketch:
Base-five numeral
_
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
ODD
60
Chapter 3
Whole Numbers
*b. Base nine
II •• III..
II •••••
Ii!!!III 11I11
III II
••
•••
III II
11I11I.
•••
• •••••••
11 •••••
• ••
III •••
IllIiiiiIIII .11I•••••••
III.
1ll1iiiiI.
•••
..11I ••••••
11I11I.
••
I!!!l •
• ••
Minimal Collection Sketch:
Base-nine numeral
c. Base ten
_
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD
DDD DDD DDD
DDD DDD DDD
DDD DDD DDD
DDD
Minimal Collection Sketch:
Base-ten numeral
_
JUST FOR FUN
The numbers 1, 2, 4, 8, 16, 32, and so on are called binary
numbers. Following 1, each number is obtained by doubling
the previous number. There are a variety of applications of
binary numbers. Many of these applications provide solutions to games and puzzles, such as the intriguing mindreading cards (shown on the next page) and the ancient game
of Nim (shown on the next page).
MIND-READING CARDS
With the five cards in Figure 1 on page 61, you can determine the age of any person who is not over 31. Which of
these cards is your age on? If you selected card 4 and card
16, then you are 20 years old. If you selected cards 1, 2, and
16, then you are 19 years old.
1. The number in the upper left-hand comer of each of
these cards is a binary number. Write your age as the
sum of the fewest possible binary numbers. On which
cards do the binary numbers that sum to your age
appear? On which cards does your age appear?
*2. Write the number 27 as the sum of the fewest number of
binary numbers. On which cards do these binary numbers occur? On which cards does 27 occur?
3. If someone chooses a number that is only on cards 1,2,
and 8, what is this number?
*4. Explain how the mind-reading cards work.
*5. The mind-reading cards can be extended to include
greater number by using more cards. The next card is
card 32 (see Figure 2). To extend this system to six cards,
more numbers must be placed on all six cards. For example, 33 = 32 + 1, so 33 must be put on card 1 as well as
on card 32. On which cards should 44 be placed?
6. Write the additional numbers that must be put on cards
1,2,4,8, 16, and 32 in Figure 2 to extend this system to
six cards. What is the greatest age on this new six-card
system?
7. Cut out the mind-reading cards from Material Card 17
and use them to intrigue your friends or students. Ask
someone to select the cards containing his or her age, or
to pick a number les than 64 and tell you which cards it
is on. Then, amaze them by revealing that number.
)