195
8
EXPLORING NUMBERS FURTHER:
STRUCTURE OF THE NUMBER SYSTEM,
NUMBER THEORY, NUMBER PATT ERNS,
& INTEGERS
TEACHING TIPS
AIMS AND SUGGESTIONS
Even after 12 or more years of school instruction, most adult students do not have a good sense
of how different types of numbers (e.g., counting
numbers, whole numbers, integers, rational numbers, real numbers) are interrelated. One aim of
chapter 8 is to help them construct a framework
for connecting these number concepts. Probe 8.1:
The Real-Number Hierarchy (pages 8-2 and 8-3 of
the Student Guide) was designed to help with this
process.
Unit 8•1: Number Theory
Key aims of Unit 8•1 are to help readers see
that the study of number theory can involve an exciting search for patterns and reflect on and better
understand important concepts included in this
topic. Specifically, Investigation 8.1: Exploring
Even-Odd Patterns (page 8-6 of the Student Guide)
can help students explore even-odd number and
arithmetic patterns, most of which will probably
be new to them. Investigation 8.2: Square-Tile
Models of Primes and Composites and Investigation 8.3: Counting Distinct Factors (pages 8-7
and 8-8 of the Student Guide, respectively) can help
them explicitly recognize concrete models of
primes and composites and explicitly define these
concepts. Investigation 8.4: A Number Trick and
Investigation 8.5: Prime Factors (pages 8-10 and
8-11 of the Student Guide, respectively) can help
students construct an explicit understanding of
prime factorization and the Fundamental Theorem
of Arithmetic. Investigation 8.6: An Example of
the Investigative Approach to Learning the Divisibility Rules (pages 8-14 and 8-15 of the Student
Guide), Investigation 8.7: Enrichment Topics in
Number Theory (e.g., perfect numbers and Goldbach's Conjecture; page 8-16 of the Student Guide),
Investigation 8.8: Exploring Squares, Square
Roots, Cubes, and Cube Roots Concretely (pages
8-17 and 8-18 of the Student Guide), and Investigation 8.9: Exploring Exponents (page 8-19 of the
Student Guide) each address the aspect of number
theory described by its title. Moreover, Investigation 8.11: Exploring Number Patterns with a Calculator (pages 8-33 and 8-34 of the Student Guide)
and the three investigations described on pages
199 to 204 of this guide can help extend the search
for patterns and foster students' appreciation for
and understanding of number theory.
Unit 8•2: Integers and Operations on
Integers
One aim of this unit is to help adult students
appreciate the importance of integers—what function they have that counting (natural) numbers do
not have. Another is to help them see that, if introduced purposefully and meaningfully (i.e., related
to everyday situations such as playing games,
credit and debit, or temperature), integers and operations on integers can be introduced as early as
the primary grades. A third aim is to help adult
students see that rules for operating on integers,
rules that most of them simply memorized and
now view as math magic, are actually shortcuts for
procedures that make sense—shortcuts that children can rediscover for themselves. Probe 8.2:
Some Questions About Integers (page 8-22 of the
Student Guide) and Investigation 8.10: Meaningful Instruction of Operations on Integers (pages
8-24 to 8-27) can help accomplish these aims.
SAMPLE LESSON PLANS
Project-Based Approach
Using the SUGGESTED ACTIVITIES (on
pages 206 and 207 of this guide) as a menu, have
individual students, pairs of students, or small
groups of about four students choose a project.
Have them share their finished project with the
class. Where applicable, involve the class in activities described in the projects.
196
Single-Activity Approach
Although numerous lessons could be devoted
to exploring number theory and integers, we will
describe only two sessions, because most instructors will not have more time for these topics.
Lesson 1. Which reader inquiries an instructor chooses to focus on will depend on which aspects of number theory, he/she feels is most important or interesting. One of many possibilities is
to focus on Investigation 8.C : A Sample of Number Patterns (pages 202 to 204 of this guide). This
reader inquiry can involve students in pattern
detection (inductive reasoning) and such interesting number-theory topics such as the Fibonacci
numbers and polygonal numbers. To explore the
former, students could be encouraged to visit the
web site cited on page 8-32 of the Student Guide.
Alternatively, a class session could be devoted
to solving and discussing a problem or two from
number theory. For example, The Case of Jason: An
Interesting Discovery About Squares and The Case of
Michael: A Close Square-Root Problem (pages 212
and 213 of this guide) would involve students in
evaluating patterns or procedures that were
discovered by elementary-level children. Such an
evaluation might help them appreciate better the
power of children's mathematical thinking and
how to implement the investigative approach and
create a spirit of inquiry—as well as reflect on two
important aspects of number theory (squares and
square roots).
Yet another alternative is watching a videotape that illustrates the investigative approach to
number-theory instruction. Deborah Ball's "Shea
Numbers" is an excellent example.* Ball's thirdgrade class had been solving problems, carefully
designed to prompt exploration of number patterns such as even and odd numbers. Shea offered
that six could be either odd or even, because three
(an odd number) groups of two or two (an even
number) groups of three could make six. Ball invited her class to consider this conjecture. The ensuing debate led to the conclusion that every
fourth number starting with 2 (2, 6, 10, 14, 18, 22
. . . ) fit this pattern. Perhaps more importantly,
*The videotape can be obtained by writing
Deborah Loewenberg Ball at the School of
Education, 4119 SEB, 610 E. University University
of Michigan, Ann Arbor, MI 48109-1259.
discussion of Shea's conjecture led the class to a
more precise definition and understanding of even
and odd numbers. Previously, the children had
known that 2, 4, 6, 8 . . . were examples of even
numbers and had settled on the inexact definition
of even numbers as those that can be split up
evenly without having to split one in half. (Note
that this definition does not specify that a collection must be evenly split in two.) While arguing
against Shea's conjecture, Ogechi offered a more
exact definition: Even numbers are the ones in
which all the items of a collection can be grouped
by twos and odd numbers are the ones where an
item is leftover. After viewing the videotape, challenge the class to solve the problem What are Shea's
Numbers? (on page 213 of this guide).
Lesson 2. Tackling Investigation 8.10: Meaningful Instruction of Operations on Integers
(pages 8-24 to 8-27 of the Student Guide) can help
students learn several useful analogies for making
sense of the addition, subtraction, multiplication,
and division of integers and how children can rediscover the formal rules for operating on integers.
Multiple-Activities Approach
Although some instructors may wish to spend
more or less time on the topic, a two-lesson unit is
illustrated below.
Lesson 1. To provide a comprehensive overview of number theory, an instructor could cover
the following portions of reader inquiries in the
Student Guide:
1. Questions 1 to 4 and 6 of Probe 8.1: The
Real-Number Hierarchy (pages 8-2 and 8-3) can
help students sort out the connections among different types of numbers. This should help them
recognize how the natural (or counting) numbers
and integers—the foci of the chapter—fit into the
real-number hierarchy. To help students better
understand the logic of Venn diagrams, include
Question 5. Many may have trouble finding or
understanding the following solution: Draw a
large circle, which is labeled Real. Within the large
circle, draw a smaller circle, labeled Rational. The
space between the two circles can be labeled Irrational, because an irrational number is a real number that is not rational. This solution is puzzling to
many students because they incorrectly conclude
it implies that irrational numbers include rational
numbers. Such students need to recognize that, in
197
Venn diagrams, hierarchical relationship are defined by concentric circles and the space inside a
circle represents a concept and the space outside it
represents "not the concept." It may help to compare the correct diagram for Question 5 with a diagram that incorrectly implies that rational numbers are a subclass of irrational numbers: A circle
labeled rational, enclosed in a circle labeled irrational, enclosed by a circle labeled real.1
2. Activity I of Investigation 8.1: Exploring
Even-Odd Patterns (page 8-6) can help students
informally define an even number as a number of
items that can be shared fairly between two people
and an odd number as a number of items where
sharing results in a leftover item ("a lonely one").
These informal definitions can help them understand why even-odd arithmetic patterns occur
(Activities III and V). For example, an odd plus
an odd is even because the leftover item (the
lonely one) for each number can be combined to
form a pair (a "couple"), which can be shared fairly
between two people. Activities II and IV can
serve as extensions or enrichment activities.
3. Activity I of Investigation 8.2 Square-Tile
Models of Primes and Composites and Investigation 8.3: Counting Distinct Factors (pages 8-7 and
8-8, respectively) can help serve to concretely and
explicitly define the difference between primes
and composites.
4. Question 1 of Questions for Reflection in
Investigation 8.4: A Number Trick and Questions
1 to 3 of Investigation 8.5: Prime Factors (pages 810 and 8-11, respectively) can provide a basis for
learning the Fundamental Theorem of Arithmetic.
Many students are curious how the magic trick introduced in the first investigation works. The second investigation illustrates how guided discovery
learning can help children satisfy their curiosity
about the trick and discover a fundamental principle about composite numbers.
5. Part II of Investigation 8.6 (pages 8-14 and
8-15) can illustrate how children can use examples
and nonexamples to induce some divisibility rules
or use logic to deduce others. Part III can help
them reflect on why divisibility rules work.
6. After establishing that a number squared
can be represented by an area model (e.g., 2 2 can
be represented by the area of a square with sides
of 2 linear units: A = S2 = 22 = the square number
4), then a class can undertake Activities II and III
of Investigation 8.8 (pages 8-17 and 8-18). By not
doing Activity IV in class and assigning the problem Estimating Cube Roots for homework, an instructor can assess transfer of learning to a moderately novel task. This can also serve to illustrate
how assessing transfer of knowledge can provide
a good indication of understanding and, thus, is an
important component of assessment.
7. Activity 1 in Part I of Investigation 8.9
(page 8-19) can help students make sense of the
following mathematical convention: Raising a
number to the zero power equals one. Activity 2 extends the pattern introduced in Part I to help them
make sense of negative exponents. It may help to
note the symmetry around 100: 10 -1 is the reciprocal of 10 1, 10-2 is the reciprocal of 10 2, and so forth.
One or more questions from Part II can help students see that the rules for operating with exponents have a sensible basis (are not merely math
magic) and that children can be guided to discover
these rules.
Lesson 2. At least some of this lesson may be
needed to finish up the reader inquiries suggested
for Lesson 1. An overview of integers can then be
provided by the following sequence of reader inquiries in the Student Guide:
1. Probe 8.2: Some Questions About Integers
(page 8-22) can serve to raise basic mathematical
and pedagogical questions about integers.
2. Activities II, III, IV, and VII of Investigation 8.10 (pages 8-24 to 8-27) can provide a meaningful overview of operations on integers. It may
be helpful to have two differently colored sets of
cubes for each group of students to experiment
with as they learn about the charged-particle analogy for operating on integers.
SAMPLE HOMEWORK ASSIGNMENTS
Lesson 1
Read: Unit 8•1 in chapter 8 of the Student Guide.
Study Group:
• Questions to Check Understanding: 3, 5, 11,
and 17 (pages 207 to 209).
• Problem: Leftover Pennies (page 212).
198
• Bonus Problem: Triangular Chips (page 213).
guess. (b) The chart does not list negative
numbers, irrational numbers, or imaginary
numbers. How could you use the chart to
identify such numbers? (c) The chart could be
expanded to include the categories of primes
and composites. Identify these types of numbers by noting in the natural-number column
-P for prime or -C for composite.
Lesson 2
Read: Unit 8•2 in chapter 8.
Study Group:
•Questions to Check Understanding: 21c, 21f, 22
and 23 (page 209).
•Writing or Journal Assignments: 7 (pages 210
and 211).
•Problem: Estimating Cube Roots (page 212).
•Bonus Problem: An Imposing Exponent (page
212).
Individual Journals: Writing or Journal Assignment
5 (page 210).
FOR FURTHER EXPLORATION
REAL-NUMBER CHART
Number
-3
7/2
56/8
0
2π
3
5
6
4i
1.
(a) Complete the Real-Number Chart below.
For examples that have not been discussed in
the Student Guide (e.g., 3 2 ), make an educated
Real
Number
Yes-P
Yes
Yes
Yes
Yes
Examine the history of the Supreme Court.
How many judges sat on each Court before
the 1930's? After Franklin D. Roosevelt tried
to pack the court with appointees to ensure his
programs would not be overturned, Congress
established that the Supreme Court would
have what number of justices? Why was this
number chosen and not, say, 10?
4.
Investigation 8.3 (on page 8-8 of the Student
Guide) was used to identify the unit (one distinct factor), the primes (two distinct factors),
and the composites (more than two distinct
factors). This activity can easily be extended
to investigate the following patterns: (a) What
kinds of numbers have exactly three distinct
factors and why is this the case? (b) Some
composites such as 6, 8, 26, and 27 can be
formed in just two ways, if the order of factors
is disregarded. (Text continued on page 205.)
Instruction on GCF and LCM illustrates how
the stage can be set for the investigative approach
to common factors or multiples, how a hundreds
square can be used to identify multiples of a number, and how a Venn diagram can be used to find
GCF and LCM.
QUESTIONS TO CONSIDER
Rational
Number
3.
Explicitly Defining and Identifying Primes
and Composites can serve to introduce students to
the Sieve of Eratosthenes and the challenging
problem of devising a model that clearly distinguishes among the unit, primes, and composites.
A Sample of Pattern Activities illustrates
number-pattern activities, including those involving triangular numbers.
Integer
When dividing natural numbers by two, we
have two categories of numbers: even and
odd. An even number is divisible by two, an
odd number leaves a remainder of one. Although dividing by two is common, it certainly is not the only number we can commonly divide by. When dividing by three, do
we still have exactly two categories of numbers: odd and even numbers? When dividing
by four, how many categories of numbers do
we have?
Investigation 8.A (page 199)
Investigation 8.C (pages 202 to 204)
Whole
Number
2.
ADDITIONAL READER INQUIRIES
Investigation 8.B (pages 200 and 201)
Natural
Number
199
Investigation 8.A: Explicitly Defining and Identifying Primes and Composites
Part I: The Sieve of Eratosthenes (◆ Identifying primes to 100 ◆ 4-8 ◆ Any number). How
Part II: Devising a Concrete Model of Prime
Numbers (◆ Fostering or evaluating an explicit
many prime numbers are there from 1 to 100? The
following procedure is a relatively simple way of
identifying the primes to 100.
understanding of prime numbers ◆ 6-8 ◆ Small
groups). Model building is an important mathematical activity. A model is essentially a set of
rules that—as accurately as possible—imitate a
phenomenon. Create a concrete model for illustrating the unit, primes, and composites. In other
words, specify a set of rules for how to use a manipulative to represent each subcategory of the
natural numbers. Then discuss your solution and
the questions below with your group or class.
1.
Put a single slash (/) through the 1 block with
a blue crayon or colored pencil. One is special:
It is the unit.
2.
Circle 2 in orange. Then cross out with red all
other numbers in the chart divisible by 2.
(Another way of saying this is: Cross out all
numbers in the chart that are multiples of 2. It
may help some children to have them count
by twos.)
3.
Circle in orange the next prime number: 3.
With red, cross out any other multiple of 3 that
has not already been crossed out. (It may help
some children to count by threes and cross out
any of these numbers not already crossed out.)
4.
Circle in orange the next prime number: 5.
With red, cross out any other multiple of 5 that
has not already been crossed out. It may help
some children to encourage them to count by
fives and cross out any numbers not already
crossed out.)
5.
Continue in this manner until all numbers are
circled in orange (the primes) or crossed out in
red (the composites). Note for what prime
number you did not have to cross out any
multiples to 100. Why was it unnecessary to
cross out any numbers for this prime? Will it
be necessary to check for multiples of the remaining primes or can you simply circle in orange all remaining numbers at this point?
Questions for Reflection
1.
A common difficulty with this activity is developing a model that is incomplete. Many
times students devise a model that distinguishes between primes and composites but
not between primes and the unit. Does your
model distinguish among all three subcategories of the natural numbers?
2.
Miss Brill had her students use square tiles to
identify the primes up to 10 by posing the
question: "What number of tiles can be arranged in two or more rows to form a rectangle?" The class noticed that some quantities
such as 4 and 6 squares could be arranged into
rows to form a rectangle, other quantities such
as 3 and 7 could not. They noted that 1 square
formed a rectangle, but it did not qualify as a
composite number because there was only one
row. Would arranging square tiles into rows
to form a rectangle correctly identify all the
primes up to 10? Why or why not?
3.
Laycock (1977) suggests that the natural numbers can be modeled by fashioning base-ten
blocks into rectangles. Excluding rotations, a
prime makes only one rectangle that is more
than one unit long and exactly one unit wide.
For example, as shown in Figure A below,
three is represented by three small blocks in a
single row. (a) Does Laycock's (1977) model
prohibit an arrangement like Figure B below?
(b) Does the model distinguish between the
unit and primes? Why or why not? (c) How
would composites be defined in this model?
Figure A
Figure B
200
Investigation 8.B: Instruction on Greatest Common Factor (GCF)
and Lowest Common Multiple (LCM)
Part I: Setting the Stage for the Investigative Approach to Common Factors or Multiples
(◆ Solving problems with common factors or multiples ◆ 3-8 ◆ Any number)
The problems below can be used as a vehicle for exploring the topic of common factors and multiples. Solve them on your own
or, preferably, with your group. Then answer
the questions that follow. Discuss your conclusions with your group or class.
■ Problem A: Square Sizes (◆ 3-8). Lester
had an 18-inch by 24-inch piece of
construction paper. He wanted to edge
the paper with equal-sized squares. How
large could he make the squares?
■ Problem B: Fair Shares (◆ 3-8). Landon
wanted everyone at his party to have the
same number of candies. He was planning on having six people at the party.
However, the Leopold twins had developed spots that morning, and it was unclear whether they had a contagious disease and would be able to attend the
party. How many candies should Landon
buy so that they could be divided equally
among either four or six children?
■ Problem C: Different Paces (◆ 3-8).
Lyndon and Lydia started running around
their house at the same time and from the
same place. Lydia ran around the house
in 3 minutes; Lyndon required 5. Assuming they could keep up this pace for 60
minutes, how many times would Lyndon
and Lydia arrive at the starting point together?
■ Problem D: The Fancy Fence (◆ 6-8).
Mr. Jung wanted to enclose his backyard
with a fence. The yard measured 65 feet
by 104 feet. He wanted to cut the top of
the fence so that it formed a series of arcs
(see figure at the top of the next column).
Mr. Jung was very finicky: He did not
want any partial arcs. What is the largest
width the arcs would have to be cut so
that each side was entirely composed of
whole arcs?
width of one arc
Fence Consisting of a Series of Arcs
■ Problem E: Coinciding Pills (◆ 6-8).
Miss Please was having a difficult first
year of teaching and was a nervous wreck.
Her doctor prescribed some red pills she
was to take every 4 hours and some blue
pills she was to take every 6 hours.
Miss Please felt especially calm whenever
she took both pills simultaneously. If she
started taking the teacher's little helpers at
the same time, how often would Miss
Please feel particularly mellow?
Questions for Reflection
1. Which of the problems above involve
finding common factors; which, common
multiples?
2. (a) What is meant by the greatest common
factor (GCF) and how does it differ from
the lowest common multiple (LCM)? (b)
Why might students confuse these terms?
3. (a) Solving which problem above involves
determining the GCF? (b) The LCM? (c)
Both?
4. The Math Book from Hell had the following
rather unrealistic fraction question: What
is the sum of 541 and 721 of a mile? Rodney
sensed that 54 x 72 (3888) was not the
lowest common denominator and that
using it would mean a lot of extra work.
(a) How could the lowest common denominator for 541 and 721 be found? (b)
What is the LCM of these fractions?
201
Investigation 8.B continued
Part II: Multiples of Numbers (◆ Identifying multiples ◆ 3-6 ◆ Any number)
Coloring multiples in on a hundreds chart can help students identify the multiples of a
number and recognize some interesting patterns. To see how and to perhaps expand your own
knowledge, answer the questions below.
1.
a.
What are the multiples of 3?
Color them in the hundreds
chart to the right.
b. Is there any pattern? If so,
describe it.
c.
How many multiples of 3 (or
any other single-digit number) are there? Hint: Imagine it is possible to keep extending the hundreds chart.
2. a.
On separate charts, identify
the multiples of other whole
numbers up to 9.
b.
Do any of these other charts
form a pattern? The multiples of what number form
columns? Parts of columns?
Slanted lines?
Parts of
slanted lines?
Part III: Using a Venn Diagram to Find GCF or LCM (◆ Number theory ◆ 6-8 ◆ Any number)
For each item below, determine the prime factors of the numbers. Then construct a Venn
diagram illustrating the union of these two sets of prime factors, noncommon (nonoverlapping)
prime factors, and common (overlapping) prime factors. Which of the following aspects of the
diagram illustrates the GCF: (a) the union of the sets of prime factors, (b) the nonoverlapping
prime factors, and (c) the intersection of the sets of prime factors? Which represents the LCM?
union of sets
Item
1.
48 and 60
2.
72 and 360
3.
36, 54, and 84
nonoverlapping
elements
Prime Factors
intersection of sets
(overlapping elements)
GCF
LCM
202
Investigation 8.C: A Sample of Pattern Activities
Part I: Some Number Patterns (◆ Number theory and inductive reasoning)
This part of the investigation includes three
activities that involve looking for number patterns.
To see what is involved and, perhaps, extend your
own understanding of number theory, try them.
Working in a group may be helpful. Discuss your
findings with your group or class.
Activity 1: Consecutive Sums† (◆ 4-8 ◆ Any
number). The number 12 can be written as the
sum of consecutive natural numbers: 3 + 4 + 5.
The number 15 can be written as the sum of consecutive natural numbers in three ways:
(a) 1 + 2 + 3 + 4 + 5; (b) 4 + 5 + 6; and (c) 7 + 8.
1.
2.
3.
can produce a litter. (Put differently, assume that
it takes one month for rabbits to reach sexual maturity and to conceive and one month for fetal development.) At the end of the second month and
every month thereafter, the original pair of rabbits
has a litter of two rabbits: one male and one female. Suppose that each new pair follows the
same reproductive pattern as its parents. How
many pairs of rabbits will there be at the beginning
of each month?
1.
Which numbers from 1 to 30 can be written as
the sum of consecutive natural numbers?
How many different ways can each of these
numbers be written in this manner? Which
numbers cannot be written as the sum of consecutive natural numbers?
In Problems with Patterns and Numbers (Shell
Centre for Mathematical Education, 1984), the
instructions direct a reader to find the numbers that can be written "as sums of consecutive whole numbers" (p. 100). Would these instructions affect the pattern you observed in
Question 1?*
What types of numbers can be written as the
sum of two consecutive natural numbers?
Solve the problem above. To make the task
manageable, note the number of pairs for only
the first seven months. To determine your answer, you may find it helps to construct a tree
diagram using the following notation:
= pair of rabbits too immature to conceive
= pair of rabbits mature enough to conceive
2.
Based on the first seven terms of the Fibonacci
sequence generated above, can you decipher
the pattern and predict the eighth, ninth, and
tenth terms in the sequence?
3.
Numbers from the Fibonacci sequence frequently appear in nature (see, e.g., Adler, 1990;
Bergamini, 1963; Pappas, 1989)
a.
Which of the following flowers have a
Fibonacci number of petals?
Activity 2: Fibonacci Numbers (◆ 6-8 ◆
Small groups or as a class). In his book Liber
Abaci, Fibonacci included a hypothetical puzzle
based upon the reproductive pattern of rabbits:
Suppose you start with a newborn male and
female rabbit. It takes two months before the pair
Trillium
† Based
on the Consecutive Sums activity described
on page 100 of Problems with Patterns and Numbers,
© 1989 by the Shell Centre for Mathematical Education, University of Nottingham.
Oriental
Dogwood
*The English use a different terminology, equating
the term whole numbers with the term natural numbers (M. Stern, personal communication, January 3,
1995).
Phlox
Daffodil
Bloodroot
203
Investigation 8.C continued
b. For each type of tree below, count how
many leaves it takes to return to the original position. Is it a Fibonacci number in
each case? For each tree how many spirals
around the stem occurred between the
bottom-most leaf and the next leaf in the
same position? Is it a Fibonacci number in
each case?
Step 1: With tiles or chips, create a model for
this series, such as that shown in Figure A.
Step 2: Move the objects assembled in Step 1
to create as many groups as possible of the next
number after the last integer in the series. For a series to 5, for example, the next number is 6 and 2 12
groups of 6 can be created (see Figure B).
Step 3: Express the results of Step 2 as a multiplication equation. For a series to 5: Total tiles =
2 12 x 6 (2 12 groups of 6).
Step 4: Consider, in turn, the sum of the first
two, three, four, six, and seven integers. Summarize your findings for each series in the Table 1 below.
Elm
Cherry
Pear
Class Activities
1.
Collect a variety of flowers growing in your
area, count the number of petals of each, and
note if the number of petals is a Fibonacci
number. (The same thing can be done with a
book on flowers, if the pictures or drawings
are sufficiently large and clear to allow counting.) Create a graphic display to illustrate the
relative frequency of Fibonacci arrays. Note
that such an activity is a good example of an
integrated mathematics-science project.
2.
A daisy head consists of two spirals that go in
opposite directions. There are 21 florets in the
clockwise spiral and 34 florets in the counter
clockwise spiral. A sunflower likewise has
two spirals of 34 and 55 florets. Do consecutive Fibonacci numbers appear in other arrangements of florets? Examine an asparagus
tip, a pine cone, and a pineapple.
Activity 3: Yet Another Look at the Sum of
Arithmetic Series (◆ 7-8 ◆ Any number). Described below are two methods for using patterns
to find the sum of an arithmetic series beginning
with 1. (See page 74 of this guide for a third
method.)
Discovery Method 1. For the sake of ease,
consider a relatively short series of integers—the
sum of the first five integers.
Step 5: In the table, what does the first factor
under total tiles represent? How can you determine this first factor if you know the last number
in a series? Can you specify a rule (devise a
formula) for finding the sum of the first n integers?
Figure A
Figure B
Table 1
Last n
2
3
4
5
6
7
Total tiles
2 12 x 6 = 15
Discovery Method 2. (a) Consider the arithmetic series 1 + 2 + 3 + 4 + 5 + 6. What if you reversed the order of the terms, placed it under the
original series, and added the two series? (b) If n is
the last number, write an algebraic expression for
determining the sum of any series 1 + 2 + 3 ... + n.
204
Investigation 8.C continued
Part II: Triangular Numbers (◆ Number theory + inductive reasoning ◆ Any number)
Polygonal numbers are sequences of numbers
named after the geometric shapes that can be used
to model the numbers (see, e.g., Figure 8.1 on page
206). This activity explores one example of polygonal numbers, the triangular numbers. To see
what is involved and, perhaps, deepen your own
understanding of this link between number and
geometry, try the investigations below yourself.
Share your findings with your group or class.
Activity 4: What is a Triangular Number?
(◆ 3-8). Set out a red checker. Below it, set out a
determine the total number of checkers in the
rows. The total number in the triangular array
would simply be one half of this.
The trick here is knowing the width and height
of the rectangle created from the sixtieth triangular
array and its duplicate. Before considering the
case of a triangular array of sixty, consider some
simpler cases. Shown below are models for the
second to fifth triangular numbers. The dark circles indicate black checkers forming the duplicate
of the triangular array (red checkers).
row with one more red checker. Continue to add
rows with one more checker than the previous
row. As this process is continued, it creates larger
and larger triangles. The total number of checkers
in each triangle (in bold print below) is called a triangular number.
second
third
1
fourth
1+2= 3
fifth
1+2+3= 6
1.
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1.
2.
3.
If you add up all the natural numbers from 1
to 8, will it form a triangular number? What
can you conclude about adding all the natural
numbers from 1 to n?
Do the triangular numbers (1, 3, 6, 10, 15 . . .)
form an arithmetic sequence? Why or why
not? Describe the pattern these numbers form
and why these numbers form the pattern.
Note that the fifth triangular number (15) is
the sum of the arithmetic series: 1 + 2 + 3 + 4 +
5 What is the sixtieth triangular number?
Activity 5: A Concrete Way of Determining
Triangular Numbers. There is a concrete way
of determining the sixtieth triangular number or
other triangular numbers. If a triangular array
were duplicated and combined with its copy to
form a rectangle, it would be relatively easy to
Complete the following chart:
Triangular
Number
second
third
fourth
fifth
Width of
rectangle
Length of
rectangle
Length x
Width
Number of
checkers in
the triangular number
2.
In each case, what do you notice about the
width and length of the rectangles? Can you
now predict the width and length of the rectangle formed by the sixtieth triangular number
and its duplicate?
3.
Let n = the ordinal position of a triangular
number (second, third, and so forth). Write a
formula for determining a triangular number
involving n checkers.
Activity 6: An Interesting Property of Triangular Numbers. Take a concrete or pictorial representation of two consecutive triangular numbers
and put them together to form a rectangle. Is there
anything special about the rectangle that is
formed? What can you conclude about summing
two consecutive triangular numbers?
205
For example, 6 is the product of 1 x 6 and 2 x 3
and, thus, has just four distinct factors. In
what way are 8 and 27 special cases of this
group of composites? What is the next special
case after 27? (c) In the table of Investigation
8.3 (page 8-8 of the Student Guide), label the
unused column on the right: Sum of Distinct
Factors. Fill in this column. What do you notice about the sum of the factors for prime
numbers? Describe a rule for determining
such sums. (d) Describe a rule for determining
the sum when a number has exactly three factors. (This question is appropriate for intermediate-level students.)
5.
6.
7.
8.
Miss Brill's initial effort to have her students
use a square-tile model to identify the primes
up to 10 was a disaster. She had instructed,
"Arrange sets of 1 to 10 squares in as many
ways as possible. How many ways can you
arrange a set of 1? A set of 2, 3, 5, or 7? A set
of 4, 6, 8, or 9?" Several groups of students noticed a pattern: A set of 1 could be arranged in
only one way; sets such as 2, 3, and 5 could be
arranged in two ways; and sets such as 4 and 6
could be arranged in more than two ways.
Most groups claimed they saw no such pattern. What about Miss Brill's instructions
might have caused confusion? How would
using a geoboard or graph paper eliminate
such a problem?
Make a "two-hundreds" chart (a hundreds
chart that runs from 101 to 200). Use the Sieve
of Eratosthenes to determine the primes from
101 to 200. (This is an enrichment activity that
some students may find interesting.)
Make a chart of two-team sports (basketball,
baseball . . .), how many players are on a team
(5 for basketball, 9 for baseball . . .), and the total number of players who play the game (10
for basketball, 18 for baseball . . .). This chart
would be useful for examining what pattern in
number theory?
(a) Is 0 a multiple of any natural numbers? (b)
When working within the natural-number
system, the terms factor and divisor are synonymous. Is this statement always true within the
whole-number system? (c) Within the wholenumber system, what is the greatest common
factor of 4 and 7? What is their least common
multiple? How does your answer compare
with the GCF and LCM when working within
the natural-number system?
9.
If the greatest common factor of two natural
numbers is 1, then these numbers are said to
be relatively prime. Any two prime numbers
such as 3 and 5 have only 1 as a common factor and are, thus, also relatively prime. Is it
possible for two composite numbers to be relatively prime? Is it possible for a prime and a
composite to be relatively prime? Briefly explain why or why not. Illustrate your explanation with several examples.
10. How many common multiples can two natural
numbers have?
11. Redo Part I of Investigation 8.A (on page 199
of this guide) on a hundreds chart with the
following modifications:
(1) Put a slash through 1 with a black crayon
or pencil.
(2) Circle 2 in red and put a red dot in the box
of all multiples of two.
(3) Circle 3 in orange and put an orange dot
in the box of all multiples of three.
(4) Circle 5 in yellow and put a yellow dot in
the box of all multiples of five.
(5) Circle 7 in green and put a green dot in the
box of all multiples of seven.
(6) Circle 11 in blue and put a blue dot in the
box of all multiples of eleven.
(7) Continue to do the same for each prime,
each time choosing a different color
(optional).
What does the completed chart tell you about
a composite number such as 24?
12. The divisibility rule for 7 is: A number is divisible by 7 if subtracting double the right-hand most
digit from the number formed by all the remaining
digits produces a difference divisible by 7. For example, for 133: 3 doubled is 6, 13 - 6 is 7 and 7
is divisible by 7. Consider now a much larger
number such as 204,512: 2 doubled is 4 and
20,451 - 4 is 20,447. Is 20,447 divisible by 7?
How could you find out—without actually dividing by long hand or with a calculator?
What does this suggest about using the divisibility rule for 7?
13. The even number 2 can be expressed as the
difference of the primes 7 and 5. Can all even
206
numbers be expressed as the difference of two
primes? Is the same true of all odd numbers?
Check the even and odd numbers up to 20 and
base your conclusion on this evidence. Hint:
The Hundreds Chart used for the Sieve of Eratosthenes can be an invaluable aid. What
kind of shortcuts can you devise using this
aid?
14. Goldbach's conjecture is that all even numbers
greater than four can be written as the sum of two
odd primes. How could this conjecture be restated so as to be true for the even numbers 4
and greater?
15. A reference listed the first 13 Fibonacci Numbers as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 141, and
233. Is this list correct? Justify your answer.
16. How many great-great-great grandparents
does a male bee have? (Note that a male has
only a mother because it develops from an unfertilized egg. Female bees have both a mother
and a father because they develop from a fertilized egg.) What number pattern does the
male bee's ancestral tree form?
17. Examine the triangular and square numbers in
Figure 8.1. Do the triangular numbers or the
square numbers form an arithmetic sequence?
Briefly describe the pattern these number sequences form.
18. Examine the pentagonal numbers in Figure 8.1. What are the next two pentagonal
numbers? Do the pentagonal numbers form
an arithmetic sequence?
Figure 8.1: Some Polygonal Numbers
A. Triangular Numbers
o
1
21. Are there any possible exceptions to the definition that n0 = 1. Justify your answer.
3
o
o o
o o o
o
o o
o o o
o o o o
6
o
o o
o o o
o o o o
o o o o o
15
10
B. Square Numbers
o
o o
o o
o o o
o o o
o o o
1
4
9
o
o
o
o
o
o
o
o
o
o
o
o
16
o
o
o
o
C. Pentagonal Numbers
o
o
o o
o o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o o o o
1
5
12
22
22. For students who have learned the properties
of exponents (specifically, xm/xn = x m-n), the
following argument can be used to demonstrate that 100 = 1:
Step 1.
10,000 104
= 1 = 103
10
10
Step 3.
100 102
=
= 101
10 101
Step 2.
1,000 103
= 1 = 102
10
10
Step 4.
10 101
=
= 100
10 101
19. To create an area model like those shown in
Figures A and B on page 8-17 of the Student
Guide, Miss Brill used round counters rather
than square tiles or a geoboard. Would this
choice of manipulative make any appreciable
difference?
20. If a cube is a geometric analogy for a number
cubed (e.g., 2 3 or 53) and a square is a geometric analogy for a number squared (e.g., 22 or
52), what is the geometric analogy for a number raised to the power of one (e.g., 21 or 51)?
What is the geometric analogy for a number
raised to the power zero (e.g., 20 or 50)?
o
o o
(a) Illustrate how the demonstration above can
be used to show what 20 and 6 0 both equal 1.
(b) Are there other ways of demonstrating that
100 = 1? Hint: Consider using the properties
of exponents you discovered in Part II of Investigation 8.9 (page 8-19 of the Student Guide).
23. In the Student Guide, it was noted that integers
represent a directed magnitude: indicate direction as well as magnitude. Do real numbers represent directed magnitudes?
SUGGESTED ACTIVITIES
1.
(a) Using references such as those listed on
page 1-36 of the Student Guide, Internet re-
207
sources such as the web sites listed on page 138 of the Student Guide, and/or television documentaries, write a brief history of number
theory. Include in your report, the social and
cultural implications or impact that the study
of number theory has wrought in the past and
how it still affects us. (b) Choose a particular
topic within number theory and examine its
historical origins and evolution.
2.
3.
4.
Design a project that would purposefully and
meaningfully engage students (a) in collecting
their own real data or data from the Internet
and (b) exploring one of the patterns discussed
in chapter 8 of the Student Guide (e.g., evenodd numbers, primes and composites, square
or cube numbers) or using scientific notation.
Using reference such as those listed on page 136 of the Student Guide and Internet resources
such as the web sites listed on page 8-32 of the
Student Guide, write a report on how and
where a pattern, such as even and odd numbers or the Fibonacci numbers, occurs in our
natural or human-made environment.
Riddles, such as the one below, can be an entertaining and challenging way to practice divisibility rules, to foster a number sense, and
to encourage mathematical reasoning. (a)
What is the solution to the divisibility riddle
below? Hint: Consider the clues in order. (b)
What kind of reasoning is involved in solving
this riddle? (c) Write your own divisibility
riddle so that there is but one answer.
layer? What is the altitude of the Salton Sea,
CA? At what point below sea level do any
components of the visible spectrum of sunlight no longer penetrate? Have the students
research these questions and summarize their
findings on a vertical number line.
7.
Collect phrases that allude to situations for
which integers are appropriate (e.g., "in the
hole" or "in the red"). Discuss their origins
and use.
8.
Devise a project that involve students in collecting their own real data or data from the
internet and exploring the integers or operations on integers.
9.
(a) Devise a spreadsheet program for listing
the first 40 Fibonacci numbers. (b) Print out
the first 40 numbers in this series.
HOMEWORK OR
ASSESSMENT
QUESTIONS TO CHECK
UNDERSTANDING
1.
The differences between the consecutive terms
in the sequence 1, 1, 2, 4, 7, 11, 16, 22, 29... consist of what type of numbers from the realnumber hierarchy?
2.
How many integers could replace N in the inequality 0 < N2 < 100 to make it true?
3.
For each of the following numbers, write the
letter(s) for the category or categories to which
it belongs. Let A = composite numbers, B =
integers, C = irrational numbers, D = natural
numbers, E = prime numbers, F = rational
numbers, and G = whole numbers.
I am a 3-digit number that is divisible by 5.
If 3 is added to me, I'm divisible by 2, 4, and 8.
If 1 is added to me, I'm divisible by 9.
What number am I?
5.
6.
Miss Brill gave her class the following three
riddles: (1) I have 3 and 6 as factors. Must I
have 18 as a factor also? (2) I am less than 100,
a multiple of 13, and divisible by 2 and 3.
Who am I? (3) I have twos and threes as my
prime factors. Am I divisible by 5? (a) Solve
the riddles above. (b) Write three riddles of
your own that involve different aspects of
number theory.
Devise an earth-science project around the
theme of elevations above and below sea level.
Pose questions such as: How high is the ozone
4.
a. -7
c.
1
2
e. 1
g.
i. 16
b. 0
d. .075
f. 2
h. 13
j. 129
Draw a concept map involving the following
concepts: (a) composite numbers, (b) integers,
(c) natural numbers, (d) positive integers, (e)
prime numbers, (f) negative integers, (g) rational
numbers, and (h) whole numbers. (Do not add
to your map any concepts that are not listed
above.) Include in your map the following examples: -2, 0, 1, 2, 4 31 , 11, 15, and 26.
208
5.
Indicate whether each statement below is—according to the Student Guide—true or false.
Change the underlined portion of false items
to make them true.
a.
Fair-sharing between two or more people
could serve as a concrete model for odd
and even numbers.
10. Miss Brill had her class use the Sieve of Eratosthenes to identify the primes up to 100. The
first four groups she checked had come up
with different answers (see Figure 8.2 on the
next page). Which group is correct and where
did the other groups err?
b. The prime factors of 1,260 are 2, 5, and
126.
11. Demonstrate or illustrate how base-ten blocks
could be used to help explain the divisibility
rules for (a) 4 and (b) 9.
c.
12. a.
The greatest common factor (GCF) of 4
and 6 is 24; the GCF of 9 and 12 is 1.
Fill in the tens and units digit so that the
resulting number is divisible by both 3
and 4:
d. The least common multiple (LCM) of 6
and 15 is 1; the LCM of 9 and 12 is 3.
e.
The number
111,111,111,111,111,111,111,111,111 is
divisible by 3; the number
999,999,999,999,999,999,996 and
888,888,888,888,888,858 are divisible by 4.
f.
In fractional form, an estimate of the
square root of 85 accurate to tenths is 9 15
.
19
g. Instruction on negative numbers should
not be introduced before grade 6 because
this concept cannot be related in a meaningful way to the everyday life of primarylevel children.
6.
Illustrate how a student could use prime factors to answer Questions 5c and 5d above.
7.
There are even-odd patterns for addition and
multiplication. Are there even-odd patterns
for subtraction and division? If so, demonstrate or illustrate how can you model the
pattern concretely with blocks?
8.
(a) In other bases, are odd numbers always
represented by the numerals 1, 3, 5, 7, 9, 11...?
(b) Are even numbers always represented by
the numerals 0, 2, 4, 6, 8, 10...?
9.
(a) In Prime-Sort, numbers such as 127, 131,
143, 211, 219, 223, 229, and 249 are printed on
3 x 5 cards. Students are challenged to sort out
the prime numbers. Which numbers above
are composites and which are primes? (b)
Annika asked, "Is -3 prime?" Her teacher encouraged her to use what she knew about
primes to determine her own answer. What is
the answer to Annika's question and why?
2
1
1
_
_
b. A whole number is divisible by each of the
numbers checked in the chart below.
What other number(s) should have been
checked?
2
√
3
√
4
5
6
7
√
8
9
10
13. (a) Circle the base-six numbers below that are
divisible by five. (b) Underline any base-six
number below that is divisible by three.
1356
2526
4336
31056
15206
14326
14. Circle the letter of any of the following that is
a multiple of 4:
(a) 132,
(b) 0,
(c) -328,
(d) 1,076,
(e) 4,442.
15. The item below appeared on the Fourth
NAEP: Less than half of the seventh graders
responded correctly; somewhat more than half
of the eleventh graders did so (Kouba et al.,
1989). (a) Answering this question would be
greatly aided by an understanding of what aspects of number theory? (b) What is the correct answer to the NAEP item?
A certain whole number is represented by B. If B is
divisible by 3, 5, and 6, then B must be divisible by
(a) 7, (b) 15, (c) 18, or (d) 14?
16. The Student Guide illustrated several ways to
help students determine the least common
multiple (LCM) and the greatest common factor (GCF). Using 9 and 12 as an example, illustrate three of these methods for finding (a) the
LCM and (b) the GCF.
209
Figure 8.2: Primes to 100 Identified By Four Groups of Students
A.
1
11
21
31
41
51
61
71
81
91
C. 1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
B.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
D.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
17. Assume that flats, longs, and cubes but not
chips (tenths of a cube) are available. (Large
cubes can’t be used in an area model because
they represent volume.) (a) Demonstrate or illustrate how base-ten blocks can be used to estimate the square root of 190 and 442 to the
nearest whole number. (b) Show how an
even more precise estimate of 190 and 442
can be determined by using the leftover
blocks and interpolation.
18. Are all square roots irrational? Briefly justify
your answer.
19. Illustrate and explain how you could convince
students that 20 = 1 makes sense.
20. To determine the answer of -3 + (-6), (a)
Garrett subtracted 3 from -6 and answered -3.
(b) Sedrick combined -3 and +6 for an answer
of +3. What might have caused each of these
errors? Briefly justify your answer.
21. Demonstrate or illustrate how you could use
counters and a charged-particles analogy to
represent (a) 4 + (-6), (b) (+4) - (+5), (c) (+4) (-5), (d) 2 x (-3), (e) -2 x (+4), and (f) -2 x (-3).
22. For the expression 7 - (-5) = +12, indicate what
each - sign would mean for (a) the chargedparticles analogy, (b) the car analogy, and
(c) the credit-debit analogy.
23. Choose an analogy other than the charged
particles model and illustrate how it could be
used to determine the answer for (a) 5 - (-2)
and (b) -1 + (-4).
WRITING OR JOURNAL
ASSIGNMENTS
1.
If Miss Brill had used yellow counters to represent 4 instead of 5 and blue counters to represent 5 instead of 7, would the number trick
described in Investigation 8.4 (page 8-10 of the
Student Guide) have worked? Why or why
not?
2.
The greatest common factor (GCF) and the
lowest common multiple (LCM) are discussed
by textbooks, but the lowest common factor
(LCF) and the greatest common multiple
(GCM) are never mentioned. Why are LCF
and GCM not considered in mathematics in-
210
struction?
3.
For Activity II of Investigation 8.7 (page 8-16
in the Student Guide), Bernice concluded that
all even (natural) numbers could be expressed
as the sum of two odd primes and that all odd
numbers could be expressed as the sum of
three odd primes. How might Bernice have
arrived at this conclusion about Goldbach's
conjecture, and why is it wrong?
4.
A student teacher asked a fifth
grader to represent with symbols a 2 by 2 display of tiles
shown to the right. 2 After a
child wrote down
, the student teacher asked, "How else can you show
what we've put out here?" When the child responded by writing:
(two squared), the
student teacher asked, "Any other ways you
can show this?" After the child indicated no,
the student teacher commented, "You could
also show it by writing this":
Pointing at the 2 2, the child remarked, "But
that's two to the second power."
(a) What does this last remark by the child
suggest about his understanding of the formal
representation of 22? (b) What should the
student teacher do to help?
5.
How would you respond to a student who
asked, "If (+10) + (-23) is an addition problem,
why do we subtract?" (a) Why is the student
confused? (b) How could the expression
above be explained in a way that makes sense
to the student?
6.
A student asks, "Why is the product of a positive number and a negative number always
negative?" How could you answer so that it
made sense to the student?
7.
Analyze the following four analogies for
adding or subtracting integers (analogies A, B,
C, and D in the next column). More specifically, in each case, indicate whether a model
(a) clearly and consistently distinguishes between + and – as operation signs and direction
signs and (b) can be used to explain all possible situations. For example, does this analogy
work for 5 + (+2), 5 + (-2) or 2 + (-5), 5 - (+2) or
2 - (+5), 5 - (-2) or 2 - (-5), -5 + (+2) or -2 + (+5),
-5 + (-2) or -2 + (-5), -5 - (+2) or -2 - (+5), and
-5 - (-2) or -2 - (-5)?
A. The confused-projectionist analogy for adding
and subtracting with integers. Some students were having difficulty understanding why you should add when subtracting
a negative integer. Mr. Findley used a
movie-projector analogy: Adding a positive number is like putting the film in correctly and going forward. Subtraction of a
positive integer is like putting the film in
correctly but going in reverse. Addition of
a negative integer is like putting the film
in backwards and going forwards. Subtraction of a negative integer is like
putting the film in backwards and going
in reverse—a trick that causes the picture
to look normal.
B. A pile-hole analogy for adding and subtracting
with integers. In the movie Stand and Deliver, Jaime Escalante explains negative
numbers as the lack of sand in a hole and
subtraction as putting sand back into the
hole (as the opposite of digging sand out
of the hole and piling it up).
C. Walking a number-line analogy for adding and
subtracting with integers.3 This analogy is
governed by the following three rules: (i)
The first addend represents the starting
point on the number line. (ii) A plus sign
indicates face right; a minus sign indicates
turn around or face left. (iii) A positive integer indicates walk forward; a negative integer, walk backwards. For example, 5 - (+2)
means start at positive five, turn around
(face left), and walk forward two spaces.
D. A difference analogy for subtracting with integers. While student teaching in an eighthgrade class, Miss Brill noticed that many
of her students had difficulty understanding the subtraction of integers. She
thought that illustrating a difference
meaning for subtraction on a number line
might be helpful. She illustrated 5 - (-2),
for example, with the following drawing:
She noted that the distance or difference between +5 and -2 was 7 spaces. Thus, +5
minus -2 was +7.
211
8.
While student teaching, Miss Brill was asked
to cover a class for Ms. Quick, who had to
leave school because of an emergency. (Her
twin sons had been caught rifling the cigarette
machine in the teacher's lounge of the high
school. They claimed they were trying to preserve the health of their teachers.) Ms. Quick
hurriedly explained, "My lesson involves introducing the addition of positive and negative integers. It's really simple. Just tell the
class: 'When you add a positive integer and a
negative integer, write the sign of the larger
number and then subtract them.' Then show
them a couple of examples, such as -8 + (+4) =
-4 and (+12) + (-9) = +3." Evaluate (a) Ms.
Quick's rule and (b) her rule-based instruction.
PROBLEMS
■ An Open-Ended Numerology Problem (◆
5-8)
Consider the following open-ended question:
Using the number code a = 1, b = 2, c = 3, and so
forth, what first names of real people would have
the same value? Identify a name that you think
would have the greatest value?
■ Copious Corn (◆ 5-8)
Mrs. Fuzz was hosting a family reunion. Her
family members had a pathological love for corn
on the cob and got terribly upset if anyone got
more than they did. Unfortunately, Mrs. Fuzz was
not sure whether all 15 family members would attend. Aunt Nel had called to say their car was
having engine trouble and that she, her husband,
and son might not be able to make the dinner.
Uncle Jes called minutes later to say they were lost
and weren't sure their family of nine would make
it either. What's the least amount of corn on the
cob Mrs. Fuzz needs to buy so that 15, 12, or 3
family members could share the corn equally?
■ The Perfect Paper-Hanger (◆ 5-8)
Mr. Meticulous, a perfectionist, wanted to
wallpaper a wall in his den. The wall was 96
inches high. There was one complication: the wall
had a 48-inch high window, 36 inches from the
floor. As compulsive as he was, Mr. Meticulous
wanted a wallpaper with a design that picked up
on top of the window exactly where the pattern
left off at the bottom of the window. Mr.
Meticulous should buy a pattern that repeats a
maximum of how many inches?
■ Beacoup Baseball Cards (◆ 5-8)
(a) Grandma Fern did not invest in land, stocks,
bonds, gold, art, stamps, coins, or any of the more
traditional securities. She invested her wealth in
baseball cards. Grandma Fern decided to divide
her collection of 100,568,121 baseball cards up
among her six grandchildren. Could she divide
her collection up so that each grandchild got the
same number and so that here were no cards left
over? (b) Grandma Fern had no sooner decided
on how to dispose of her accumulated wealth
when she learned that her youngest daughter
Bernice had twins. Could she divide her collection
up evenly among eight grandchildren? (c) A few
moments later a dazed and crazed Bernice called
back to say she actually had triplets. Grandma
Fern decided to buy just enough more cards so
that her collection could be divided by nine. How
many cards does she need to buy?
■ Some Differences † (◆ 5-8)
A total of 1988 different whole numbers are
written down. The difference between the number
of even numbers recorded and the number of odd
numbers recorded can be any of the following except: (a) 0 (b) 1 (c) 2 (d) 600 (e) 994 (f) 1988.
■ Multidigit Prime Factor (◆ 7-8)
(a) What is the first number with a prime factor greater than 10? (b) What is the first three-digit
number with a prime factor greater than 10? (c)
What is the first three-digit number with a twodigit prime factor?
■ Oddly Enough (◆ 7-8)
(a) What is the 1991st odd whole number? (b)
What is the sum of the first 100 odd whole numbers?
■ Divisibility Rules for Other Bases (◆ 7-8)
(a) Devise divisibility rules for 2, 3, 4, and 5 in
base six. (Consider what you know: the divisibility rules in base 10. Might any of these rules be
† Based on a question from the 1991-92 Sixth Grade New
York Math League Contest authored by Steven R.
Conrad and Daniel Flegler.
212
applicable to the new task at hand? Consider the
rationale for the base-10 divisibility rules. Might
any of these rationales be applicable to your task?)
(b) Compare the base-six divisibility rules with
those for base 10. What parallels do you notice?
What parallels might there be to other base systems? For example, compare the divisibility rule
for 5 in base 6 with that for 9 in base 10. What
conclusion can you draw? List the first five numbers in base 5 divisible by 4. Does this support or
weaken your conclusion?
■ Leftover Pennies (◆ 7-8)
At the end of the year, a club had $5.29 left in
its treasury. If the money was distributed equally
among the club members with every member receiving five coins (the same combination of five
coins), what was the total number of pennies distributed?
■ Estimating Cube Roots (◆ 7-8)
After her class figured out how to estimate the
square roots of numbers, Miss Brill proposed the
following small-group homework assignment:
Use a calculator to help you estimate the cube root
of 550 to thousandths. Show your work. Richard
voiced the feelings of many students when he
protested: "This isn't fair, Miss Brill. We've never
studied how to estimate cube roots." Miss Brill,
unmoved, indicated it was a thinking problem. (a)
What is the answer to Miss Brill's assignment? Be
sure to show how you obtained your answer. If
available, use the cube root function of your calculator to check your estimate. (b) Why did Miss
Brill apparently feel justified in giving the class an
assignment that involved something they had not
been taught? Was she just covering up a slipup or
was there a good reason for the test item?
■ A Generous Gift Giver (◆ 7-8)
In the song "The Twelve Days of Christmas," a
gift giver gives a partridge in a pear tree on one
day; two turtle doves and a partridge in a pear tree
on a second day; three French hens, two turtle
doves, and a partridge in a pear tree on third day;
and so forth. How many gifts did the gift giver
give on the twelfth day? How many gifts did he
give total for the twelve days? The number of gifts
given on each day of the twelve days is what kind
of number?
■ The Case of Jason: An Interesting
Discovery About Squares4 (◆ 7-8)
Jason's fourth-grade teacher encouraged her
students to explore number patterns on their own.
At least once a week, groups of four children met
and decided on a problem to tackle. Jason's group
became particularly interested in square numbers
after they discovered that only certain numbers of
blocks will form a square. Using interlocking
blocks of different colors, they constructed an expanding square shown below.
Square root (Side: Number of blocks comprising a side)
12
11
10
9
8
7
6
5
4
3
2
■ An Imposing Exponent (◆ 7-8)
1
1
Miss Bright posed the following problem:
What is the units digit for the answer of 31990? You
may use your calculators.
"Great tofu globs, Miss Bright," complained
Richard. "Even if we use a calculator, it will still
take us a million years to multiply out." When
Miss Bright smiled slyly, Richard concluded,
"There's got to be an easier way." Without multiplying three as a factor 1990 times, can you determine what the ones digit will be?
4
9
16
25
36
49
64
81
100 121 144
Square
(Area: number of blocks in a color band plus those within the band)
a.
All the members of Jason's group had
recorded the squares for the numbers 1 to
35 in their notebooks. Jason took his notebook home over Thanksgiving break so
that he could extend his list and look for
patterns. Jason discovered a pattern that
allowed him to extend his table to 100
without multiplying. He found that the
213
square for a number could be determined
by adding the number, the number before
it in the counting sequence, and the square
of this previous number. For example, to
find the square of 5, add 5 + 4 + 16: 25.
Why does Jason's algorithm work?
two columns of figures. Is the estimated
square accurate when rounded to the
thousandths place? Hundredths place?
Tenths place?
Number
3
5
9
15
18
b. Jason also discovered that if you squared
any number ending with 1 (e.g., 11 or 31),
the squares would end with 1. If you
squared any number ending with 2 (e.g.,
22 or 92), the square would end with 4.
Why? What will the square of 43 and 83
end in? What about 19, 59, and 99?
e.
Michael announced that a better estimate of
the square root of 20 was 4 49 .
a.
Is 4 49 a better estimate of the square root
of 20 than is 4.4?
1
Square roots
of perfect
squares:
1
Estimated square
roots of nonperfect
squares:
c.
3
4
5
9
2
12
3
In what way is Michael's method similar
to using base-ten blocks and interpolation
to estimate the square root of 20 (or other
numbers)?
■ Triangular Chips (◆ 7-8)
Using chips, creInterior
ate a triangle with chip
four chips per side.
Fill in the interior
with enough chips so
all the chips in the
array maintain contact (see diagram above). (a)
How many chips will you need to create a triangle with 100 chips on a side? (b) How many
interior chips will there be in such a model?
■ What are Shea's Numbers? (◆ 7-8+)
b. The child had apparently discovered an
algorithm for estimating square roots. Below are some examples of estimates generated by this algorithm. Note that the
perfect squares and their square roots are
show in boldface. What algorithm did
Michael use to generate the estimated
square roots of nonperfect squares?
Number:
Decimal equivalent of
estimated square root
d. Use the algorithm induced in b to estimate
the square root of 90. To what place is this
estimate accurate?
■ The Case of Michael: A Close SquareRoot Problem4 (◆ 7-8)
Jason presented his discoveries to the class.
His work excited other groups to see what they
could discover about squares and square roots.
One group discovered that all whole numbers
have square roots and that square roots do not
have to be positive integers. Using base-ten
blocks, they estimated the square root of 20 to be
4.4. That is, they showed that a square with sides
of 4.4 blocks would be about 20 blocks.
Square root
(to hundredths)
15
3
21
5
16
18
20
4
36
7
25
5
42
9
44
9
Use a calculator to determine the square
roots of 3, 5, 15, 18, and 20 to the thousandths place. Convert the mixed number
estimates for these numbers to decimals.
Fill in the table below and compare the
Deborah Ball's third-grade class was exploring
number patterns, including even and odd numbers. Shea offered that six could be either odd or
even, because it could be represented by three (an
odd number) groups of two or two (an even number) groups of three. The ensuing class discussion
led to the conclusion that starting with 2, every
fourth number fit this pattern (see the table below). For example, 10 can be viewed as five (an
odd number) groups of two or two (an even number) groups of five. What is the pattern underlying "Shea's Numbers": 2, 6, 10, 14, 18, 22 . . . ?
o
1
o
9
o
17
o/e
2
o/e
10
o/e
18
o
3
o
11
o
19
e
4
e
12
e
20
o
5
o
13
o
21
o/e
6
o/e
14
o/e
22
o
7
o
15
o
23
e
8
e
16
e
24
214
ANSWER KEY for Student
Guide
legs (e.g., 5:
1.
A rational number with 0 in the denominator
would be undefined.
2.
Yes, because 3 and 4 can be put in the form of
a
, where a and b are integers, and b ≠ 0
b
3.
a.
3
1
and
4
1
).
1.
(a) Yes; 124 = 6, and six items can be evenly
shared between two people. (b) No; 134 = 7;
(c) No, 125 also equals 7. (d) Yes; 135 = 8.
2.
See page 34 of this guide (the key for Questions 2 & 3, Part I of Investigation 1.4).
Activity III
Yes, because it can be put in the form of
0
1
1.
and meets all the criteria outlined in the
previous paragraph.
b. Yes,
3.5
2
=
7
4
and, thus, meets all the
defining criteria for a rational number.
4.
).
Activity II
Key for Probe 8.1 (pages 8-2 and 8-3)
(namely
and 7:
With any luck, students will discover the patterns and pattern justifications illustrated below for themselves. Many will make these
discoveries in short order. Others will need
some support to do so.
Odd Plus Odd is Even
(c), (d) and (e) (the counting numbers 1, 3, and
10) only.
5.
A block (odd block out)
in each set
(odd number)
is left unpaired.
Odd blocks out can
now be paired with
another block.
Odd Plus Even is Odd
Questions for Reflection
1.
Yes, the imaginary numbers.
2.
If the hypotenuse of a right triangle (labeled c)
with two equal sides of 1 unit (labeled a and b)
were placed on a number line, then the hypotenuse would span the distance from 0 to
2 (c2 = a 2 + b2 = 1 2 + 12 = 2; thus, the length
of c = 2 ).
Key for Investigation 8.1 (page 8-6)
Activity I
1.
2.
(c) Yes, Duena’s conjecture can be modified to
read:
Square numbers can form a box
[rectangle] consisting of two rows, but odd
number can’t.
With the exception of 1, an odd number of
tiles or cubes can form an L with equal-sized
Odd block out still
cannot be paired
with another block.
Even Plus Even is Even
All blocks in each set can be paired.
2.
The rectangle = even and nonrectangle = odd
model is a powerful visual analogy for oddeven arithmetic patterns.
(a) odd + odd = even:
+
➞
(b) odd + even = odd:
+
➞
(c) even + even = even:
+
➞
215
Activity IV
a.
As the examples below show, the sum of consecutive odd natural numbers beginning with
one is a perfect square. Note that the number
of terms in each arithmetic series below is the
square root of the sum of the terms.
out and thought of as: (3 - 2) + 3 + (3 + 2) = 3 +
3 + 3 or 3 x 3 or 3 2. This can be concretely illustrated with blocks as shown in Figure 8.3:
Activity V
odd (3) x odd = odd:
➞
odd (3) x even = even:
First two: 1 + 3 = 4
First three: 1 + 3 + 5 = 9
First four: 1 + 3 + 5 + 7 = 16
First five: 1 + 3 + 5 + 7 + 9 = 25
➞
b. The sum of the first eight consecutive odd
natural numbers (1 + 3 + 5 + 7 + 9 + 11 + 13 +
15) is the square of eight: 64. The sum of the
first twelve consecutive odd numbers is the
square of 12 or 144.
c.
"Why does this work?" Consider 1 + 3. One is
one less than 2 and 3 is one more than 2. Thus,
1 + 3 = (2 - 1) + (2 + 1). The minus one and
plus one cancel, leaving two twos: 2 + 2,
which is equal to 2 x 2, which can be represented as 22. Likewise, 1 + 3 + 5 can be evened
even (2) x even = even:
➞
Key for Investigation 8.2 (page 8-7)
Activity II
1.
The rectangular numbers are composites such
as 6 (see Figure 8.4 below), 9, 12, and 15.
2.
The unit (1) or primes (e.g., 2 and 3) cannot be
modeled in two or more equal rows of more
than one object in a row. Four, a composite
number, can be.
Figure 8.3: Why Adding Consecutive Odd Numbers Equals a Square
(1 + 3 + 5 = 3 + 3 + 3)
Using the blocks, we can also see how the squares are built up.
(2 x 2 = 22 = 4)
1 + 3:
(3 x 3 = 32 = 9)
1 + 3 + 5:
Figure 8.4: Rectangular Area Model of Six Using a Geoboard Model*
6=6x1
6=3x2
6=2x3
6=1x6
*Graph or dot paper could be used instead of geoboards.
216
Activity III
5.
Primes such as 13 can only be arranged in two
equal rows of items (1 x 13 and 13 x 1), while composites such as 12 can be arranged in more than
two equal rows (1 x 12, 12 x 1, 2 x 6, 6 x 2, 3 x 4,
and 4 x 3).
Key for Investigation 8.3 (page 8-8)
1.
There are several ways to go about factoring a
number. One way is to begin with a readily
apparent factor. In the case of 320, 10 is an
easy factor. Other candidates for easy factors
include 4, 8, 20, and 160. Another way to approach factoring is to begin with the smallest
prime 2 and work up to larger primes until the
factoring is completed. These two methods
are illustrated below:
320
The number 1 is unlike any other number, be-
cause it has only one distinct factor.
2
3.
10
2
(a) The numbers 2, 3, 5, and 19 are all prime:
have only two distinct factors (1 and themselves).
3.
They have more than two distinct factors.
This theorem also explains why primes are
defined in such a way as to exclude 1. If 1
were defined as a prime, then the prime factorization of 8, for instance, would yield infinitely
many sets of prime factors, including
1 • 2 • 2 • 2 and 1 • 1 • 1 • 1 • 2 • 2 • 2, not
simply the unique set: 2 • 2 • 2. Thus, the
Fundamental Theorem of Arithmetic would
not be true. Basically, then, 1 is not included
in the definition of primes so that we can have
The Fundamental Theorem of Arithmetic.
16
2
8
2
4
2
2
Starting with an easy identifiable
factor, in this case, 10
320
Any composite number can be factored into a
set of primes (e.g., the prime factors of 39 are 3
and 13). Even when there is more than one
way to factor a composite, the result is the
same set of primes. For example, 48 can be
factored into 2 • 24, 3 • 16, 4 • 12, or 6 • 8, and
each results in the same set of prime factors:
2 • 2 • 2 • 2 • 3.
Moreover, the Fundamental Theorem of Arithmetic states that, except for order, each composite
number can be factored into a unique set of primes.
For example, 12 can be reduced to a single set
of primes (2 • 2 • 3) that is different than the
set of prime factors for any other composite
numbers. Writing a composite number as the
product of all of its prime factors (divisors) is
called prime factorization. This should make
clear to students how Miss Math Magic determined the counters chosen in the number
trick described in Investigation 8.4 (on page 810 of the Student Guide).
5
2
Key for Investigation 8.5 (page 8-11)
2.
32
2
160
2
2
80
2
40
20
2
2
10
5
Starting with the smallest prime and
checking out progressively larger primes
Note that the second method is highly systematic and results in listing the prime factors in
order of size.
Key for Investigation 8.6 (pages 8-14 and
8-15)
Part III
1.
We can disregard the tens, hundreds, and
thousands, because all are divisible by two
(10 ÷ 2 = 5, 100 ÷ 2 = 50, 1000 ÷ 2 = 500). This
is true whether an even or an odd digit is in
these places. Consider, for example, the concrete model of 1336. The large cube can be
traded for 10 flats, which can be evenly divided into two groups. The three flats can be
traded in for 30 longs, which can be evenly divided into two groups of 15 longs. The three
longs can be traded in for 30 cubes, which can
be evenly divided into two groups of 15 cubes
each.
2.
With 126, a hundred (flat) can be divided up
into 33 groups of 3 cubes, leaving one cube leftover. Each ten (long) can each be divided up
217
into 3 groups of 3 cubes, leaving one cube leftover. Because all the leftover cubes plus those
representing the ones digit form a collection
divisible by 3 (1 + 2 + 6 = 9), the number 126 is
divisible by 3.
100 ÷ 3 = 33 r 1
100 ÷ 9 = 11 r 1
10 ÷ 3 = 3 r 1
(for each long)
1
10 ÷ 9 = 1 r 1
(twice)
+
2
+
6
Key for Investigation 8.7 (page 8-16)
Activity I: Perfect, Deficient, and Abundant
Numbers
1
3.
5.
6.
2
+
6
(b) We can disregard hundreds, thousands,
and so forth because all are divisible by four
(100 ÷ 4 = 25 and 1000 ÷ 4 = 25). Consider the
concrete model for 1236, the large cube can be
traded in for 100 longs which can be divided
into 25 groups of four. The two flats can each
be traded in for 100 cubes, which can be
divided into 25 groups of four. The three
longs can each be traded in for 30 cubes and
combined with six cubes. The total (36) can be
divided evenly into 9 groups of four.
250 groups
of 4
4.
+
25 groups of four
each
9 groups of
four
We can disregard all but the ones because
tens, hundreds, thousands, and so forth are all
evenly divisible by five. Thus, if the number
of units is divisible by five, the number as a
whole is divisible by five.
We can disregard digits beyond the hundreds
place because 1000, 10,000, 100,000, and so
forth are all divisible by eight.
Eleven groups of 9 can be made from the hundred (flat), leaving a single cube leftover.
Likewise, each ten (long) can make a group of
9 (cubes), leaving a single item leftover. If the
leftover items plus the items representing the
ones digit form another group of nine, then
the number is divisible by 9.
1.
Between 1 and 30 there are only two perfect
numbers: 6 and 28. Perfect numbers appear to
be rare.
2.
The girl may have reasoned that zero is divisible by any whole number except 0 (Premise 1).
Given the definition of a proper divisor
(Premise 2), it logically follows that zero has
an infinite number of proper divisors.
3.
The number 1 is divisible by itself only and
thus, has no proper divisors.
4.
(a) Yes. Prime numbers have only 1 and
themselves as divisors and, thus, only 1 as a
proper divisor. (b) A deficient number.
5.
The number 496 is the third perfect number.
6.
There are no odd perfect numbers up to 30.
Indeed, to date, none has been found beyond
30 either. It remains unknown whether there
are any odd perfect numbers.
Activity II: Goldbach's Conjectures
All even numbers six or greater can be written
as the sum of two odd primes. All odd numbers
nine or greater can be written as the sum of three
odd primes.
Key for Investigation 8.8 (pages 8-17 and
8-18)
Activity II
5.
Let a flat = 1 square unit, a long = .1 square
units, and a cube = .01 square units. The
number (area) 20 would be represented by 20
flats. Arranging the flats to make the largest
218
square possible produces a 4 by 4 array of 16
flats. To extend the square, trade in the remaining flats for longs and cubes. This allows
you to construct the model illustrated in Figure 8.5 below. Note that the answer (the unknown side of the square) is represented by
the length of 4 flats and width of 4 longs or
about 4.4 linear units.
6.
7.
As the last diagram in Figure 8.5 shows, the
leftover blocks (6 longs + 4 cubes or 64 cubes)
are not enough to make a larger (a 4.5 x 4.5
linear units) square, which would require an
additional 8 longs and 9 cubes or 89 cubes.
Thus, the leftover cubes are 64
of what is need89
ed to add another 0.1 linear units to each side
of the existing (4.4 x 4.4 linear units) square.
As 64
= 0.719101 (according to a TI Math Ex89
plorer), the leftover blocks constitute 0.719101 x
0.1 or 0.0719101 lin. units. Thus, a more accurate estimate is 4.4 + 0.719101 or 4.4719101.
Let a flat = 100 square units, a long = 10 square
units, and a cube = 1 square units. An area of
150 square units would be represented by a
flat (100 square units) plus five longs (5 x 10
square units or 50 square units). The largest
possible square that can be constructed from
these blocks is shown to the upper right.
Thus, the square root of 150 to the nearest
whole number is 12 (the length of 1 flat + the
width of 2 longs). To get an answer accurate
to tenths, interpolation is required. Because it
would take another 25 blocks to complete the
next largest square and we already have 6 of
those blocks, a more precise estimate of
150 is 12 and 6/25ths or 12.24; 12.2 rounded
to tenths. (The actual square root of 150 is
12.247449..., 12.2 rounded to tenths.)
Figure 8.5: Area Model of
12
12
Activity III
1.
Box 8.1 (on the next page) illustrates how a
calculator can be used to estimate square
roots. For another method for estimating
square roots, see The Case of Michael: A CloseSquare Root Problem on page 213 of this guide.
2.
Students were once required to memorize the
algorithm for determining square roots. The
advent of inexpensive hand-held calculators
made learning this laborious algorithm
unnecessary, and it is no longer a part of
recommended syllabi. It is useful, though,
that students be able to estimate square roots
for several reasons: to foster mathematical
thinking, in general, and the strategy of interpolating (gauging an unknown value between
two known values), in particular; to deepen
students understanding of square roots; and to
learn a method for checking calculated answers.
Activity IV
2.
(a) 23 can be modeled by a 2-cube long by 2-
20 Using Base-Ten Blocks
4.4
An area of 20 sq. units can
be represented by 20 flats,
16 of which can be arranged
in a 4 x 4 square.
The remaining 4 flats can be traded for
40 longs, 2 of which can then be traded
for 20 cubes.
Leftover:
219
Box 8.1: Estimating Roots
Miss Brill began her lesson on estimating the square roots of nonperfect squares with a question:
"How could you determine the square root of 20 if you didn't have a calculator with a square-root
function?"
"You could estimate it I suppose," responded Helen.
"How would you ever estimate the square root of 20?" commented Bob skeptically.
Rodney, in a flash of insight, answered excitedly: "Well, you know it's got to be greater than 4
because 20 is bigger than 16 [the square of 4]. But it's got to be less than 5 because 20 is smaller than 25
[the square of 5]." To drive home his point Rodney drew:
4
16
?
20
5
25
4
?
5
"It's a little less than 4.5 because 4.5 times 4.5 is 20.25," interjected LeMar.
"If we round our answer to tenths, would 4.5 or 4.4 be a better estimate?" asked Miss Brill.
"Well, 4.4 x 4.4 is 19.36," noted LeMar. "So 4.5 would be the closer estimate."
"What if I wanted an estimate accurate to hundredths?" asked Miss Brill devilishly. "You may use
your calculators if you wish—except for the square-root function."
Knowing that the product of 4.5 x 4.5 was greater than 20, LeMar tried 4.48 x 4.48 and got 20.07. He
tried 4.47 x 4.47 and got 19.98. Deciding the latter answer was closer to 20, LeMar settled on 4.47 as his
estimate of the square root.
cube wide by 2-cube high cube. (b) The length
of each side of the cube. (c) The 8 represents
the volume.
2.
1
1
1
1
1
1
1
1
36 ÷ 34 = (3•3• 3 • 3 • 3 • 3 ) ÷ ( 3 • 3 • 3 • 3 )
2
2
= (3•3) ÷ 1 = 3 ÷ 1 = 3 (Keep the common
base and subtract the exponents.)
Key for Investigation 8.9 (page 8-19)
3.
(24)3 = 2•2•2•2 three times or 212.
Part II
4.
23•53 = (2•2•2)(5•5•5) = (2•5)(2•5)(2•5) = 103
1.
Key for Probe 8.2 (page 8-22)
(a) A teacher can encourage children to discover the rule for multiplying exponentials
with a common base. After posing a question
like "What's 2 4 x 25?", a teacher could urge students to use their existing knowledge: 24 represents what? (2 x 2 x 2 x 2); 25 represents
what? (2 x 2 x 2 x 2 x 2). Therefore, 24 x 2 5
represents what? (2 x 2 x 2 x 2) x (2 x 2 x 2 x 2
x 2) or 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 29. It
may help to have them examine other relatively simple cases, such as 32 x 33 = (3 x 3) x
(3 x 3 x 3) = 3 x 3 x 3 x 3 x 3 = 35. With any
luck this will help students notice the following pattern: Raise the common base to a power
equal to the sum of the exponents.
1.
(a) Alexi's experience suggests that games can
help children understand the need for negative numbers and can provide a purposeful
context for introducing the topic. (b) Elevation
above and below sea level; time before and after a rocket launch or the beginning of a military maneuver. (c) Integers indicate direction
as well as magnitude.
2.
No, the student teacher is incorrectly interpreting the integer signs (e.g., -6 and +7) as implying an operation rather a direction. Teachers
should help their charges distinguish between
these two meanings of + and -.
220
3.
Get to it. There is good reason to believe that
second-grade students can learn about integers in a meaningful and purposeful manner.
formed into 5 x (-3), which can be viewed as -3
added five times.
3.
Key for Investigation 8.10 (pages 8-24 to
8-27)
Activity V
Activity I
2.
Word
Problem
A
B
C
D
E
F
G
Informal
Format
12 - 7 = 5
5 + 7 = 12
12 - 17 = -5
-5 - 7 = -12
-12 - 10 = -22
-22 + 20 = -2
-2 + 7 = 5
Formal
Format
12 + (-7) = 5
5 - (-7) = 12
12 + (-17) = -5
-5 + (-7) = -12
-12 - (+10) = -22
-22 + (+20) = -2
-2 - (-7) = 5
Activity II
(a) Add three groups of positive two (+6). (b)
Add three groups of negative two (-6). (c)
Take away three groups of +2 (-6). (d) Take
away three groups of -2 (+6).
Activity VI
1.
If need be, provide students the following
hint: Consider -18 ÷ -3 as a missing secondfactor multiplication expression and -18 ÷ 6 as
a missing first-factor multiplication expression. Then apply the rate analogy for integer
multiplication.
2.
If need be, provide students the following
hints: (a) How many groups of negative three
do I need to add or subtract to a
neutral box to give it a net charge of -6?
(b) How many groups of positive three
(
) do I need to add or subtract to a
neutral box to give it a net charge of -6?
(c) How many groups of negative three
do I need to add or subtract to a
neutral box to give it a net charge of +6?
Expressions such as (+3) - (-2) require considering the neutral charges in the box. The expression calls for taking away two negative charges
from a box that starts with a net charge of positive
three (
). This can be done by considering
two of the infinite number of neutral charges
(
(a) No. (b) It could be viewed as the repeated
subtraction: -5 subtracted three times.
) in the box.
(1) Represent the starting net charge of +3.
(2) Represent two neutral charges so that two
negative changes can be removed.
(3) Count the remaining charges: +5.
Key for Investigation 8.11 (pages 8-33 and
8-34)
Activity II: Nines Gone Wild
This activity illustrates how using patterns can
make an apparently overwhelming task manageable or even easy.
Step 1: Start
Step 2: Change
Step 3: Outcome
Activity III
1.
Yes; the direction sign indicates whether the
car is driving forward or backward (in reverse), and the operation sign indicates what
direction the car is pointing.
2.
Yes.
Activity IV
1.
Yes.
2.
Yes; by applying the commutative law of
multiplication, the expression can be trans-
1.
By creating a table and using a calculator to
compute the products, students should be able
to decipher a useful pattern quickly:
1 x 99,999 = 99,999
2 x 99,999 = 199,998
3 x 99,999 = 299,997
4 x 99,999 = 399,996
Note that the first and last digit of the product
in each case above combine to form the product of the multiplier and 9 (compare, e.g., the
first and last digits of the product of 2 x 99,999,
199,998, with the product of 2 x 9, 18). In between the first and last digit are four nines:
one less than in 99,999.
221
3.
The first step of the algorithm is shown below.
8
999,999,999,999,999
x9
1
The second step:
88
999,999,999,999,999
x9
91
Note that each successive multiplication will
yield the same product: 81. Because the 1 is
added to the 8 carried over from the previous
step, the sum is always 9. The 8 is carried to
the next column. This process repeats itself
until there are no more digits to multiply.
888 888 888 888 88
999,999,999,999,999
x9
8,999,999,999,999,991
Activity III: A Square Root Too Far
Perhaps surprisingly, repeatedly hitting the
square-root key on a calculator eventually reduces
any number entered to 1. Is the twenty-fourth
square root of 2 or the twenty-fifth square root of 3
actually 1? No, it is an approximation. The calculator has rounded off the twenty-fifth root. Graphing can illustrate that successive square roots approach the value 1. The activity informally introduces the concept of limits: approaching but never
actually achieving a value. One problem with
graphing the square root of 3 is the inaccuracy due
to the need to round off values. With a graph, the
square root of 3 reaches—for all practical purposes—1 fairly quickly. (The graph is a visual
analogy of what the calculator eventually does.)
Activity IV: Patterns in the Fibonacci Numbers
Multiples of some of first five primes repeat.
For example, every third Fibonacci number is a
multiple of 2. Multiples of other numbers in the
sequence also repeat.
Activity V: Dividing by Eleven Made Easy
a.
1
If a child remembers that 11 is .09 infinitely
repeating, then he or she can easily figure out
the decimal equivalents of n by multiplying
11
—
.09 by n.
—
3
11
= .272727 27
—
4
11
= .363636 36
1
11
= .090909 09
2
11
= .181818 18
—
—
b. The pattern holds for improper fractions also.
14
For example, 11 can be converted into the
3
mixed fraction 111. The decimal equivalent is
—
—
1 + (.09 x 3) or 1.27.