Download Activity Assignement 4.1 Number Theory

CHAPTER
Number Theory
Number theory offers many rich opportunities for explorations that are interesting, enjoyable,
and useful.
... Challenging but accessible problems from number theory can be easily formulated and
explored by students. For example, building rectangular arrays with a set of tile can stimulate
questions about divisibility and prime, composite, square, even, and odd numbers .... 1
Activity Set
4.1
Virtual
Manipulatives
PURPOSE
..• crB.J.J
•
I
•
•
MODELS FOR EVEN NUMBERS, ODD NUMBERS,
FACTORS, AND PRIMES
•
To use models to provide visual images of basic concepts of number theory .
MATERIALS
Color Tiles from the Manipulative Kit or from the Virtual Manipulatives.
www.mhhe.comlbbn
INTRODUCTION
Some problems in number theory are simple enough for children to understand yet are unsolvable by mathematicians. Maybe that is why this branch of mathematics bas intrigued so many
people, novices and professionals alike, for over 2000 years. For example, is it true that every
even number greater than 2 can be expressed as the sum of two prime numbers? It is true for the
first few even numbers:
4=2+2
8=3+5
6=3+3
10
=
5
+5
12
=
5
+7
14
= 7 +7
However, mathematicians bave not been able to prove it is true for all even numbers greater
than 2.
The ideas of odd, even, factors, and primes are basic concepts of number theory. In this
activity set, these ideas will be given geometric form to show that visual images can be associated
with them.
ICurriclllllm
Mathematics,
and Evaluation
1989): 9 J -93.
Standards
for School Mathematics
(Reston,
VA: National
Council
of Teachers
of
95
Activity Set 4.1
97
Models for Even Numbers, Odd Numbers, Factors, and Primes
Even and Odd Numbers
1. The nonzero whole numbers (counting numbers) can be represented geometrically in many
different ways. Here the first five consecutive numbers are represented a a sequence of
tile figures, Extend the sequence by drawing the figures for the numbers 6, 7, and 8.
2
3
4
5
6
8
7
Describe, in your own words, how the figures for the even counting numbers differ from
those for the odd counting numbers in the sequence above.
2. Here is a tile sequence for the first four even numbers.
tst
2d
3d
4th
2
4
6
8
7th
a. Draw the figure for the seventh even number above. What is the seventh even number?
Seventh even number
_
*b. Describe in words what the figure for the 125th even number would look like. What is
the 125th even number?
125th even number
_
c. Describe in words what the figure for the nth even number would look like. Write a
mathematical expression for the nth even number.
nth even number
_
98
Chapter 4
Number Theory
3. This tile sequence represents the first four odd numbers.
1st
2d
3d
4th
3
5
7
5th
8th
a. Draw the figures for the fifth and eighth odd numbers. What is the fifth odd number?
What is the eighth odd number?
5th odd number
_
8th odd number
_
*b. Describe in words how to draw the figure representing the 15th odd number. What is the
15th odd number?
15th odd number
_
c. Determine the 50th, lOOth, and nth odd numbers by imagining what their figures would
look like, based on the model shown above.
50th odd number
_
100th odd number
_
nth odd number
_
*d. When the odd numbers, beginning with 1, are arranged in consecutive order, 7 is in the
fourth position and 11 is in the sixth position. In what position is 79? 117? (Hint: Think
about the tile figures for these numbers. You may wish to draw a rough sketch of the
figures.)
79 is in the
position.
117 is in the
position.
4. The following diagram illustrates that when an even number is added to an odd number,
the sum is an odd number. Determine the oddness or evenness of each of the sums and differences below by drawing similar diagrams.
----
Even number
Odd number
a. The sum of any two even numbers
*b. The sum of any two odd numbers
-
Even + Odd number = Odd number
Activity Set 4.1
Models for Even Numbers, Odd Numbers, Factors, and Primes
99
c. The sum of any two consecutive counting numbers
d. The sum of any three consecutive counting numbers
*e. The difference of any two odd numbers
f. The sum of any three odd numbers and two even numbers
5. The tiles for the figures in the tile sequence for consecutive odd numbers can be rearranged
as follows:
Original sequence
D~~~~
R"""9'dI;"< D
Repositioned
~
~
L
tiles
a. The L-shaped figures for the first five consecutive odd numbers can be pushed together
to form a square. What does this tell you about the sum of the first five consecutive odd
numbers?
100
Chapter 4
Number Theory
*b. Visualize the L-shaped figures for the first 10 consecutive odd numbers. What size
square can be formed from these figures? What is the sum of the first 10 consecutive
odd numbers?
c. Consider the sum of all odd numbers from 1 to 79.
1
+ 3 + 5 + 7 + ... + 77 + 79
How many numbers are in this sum? Determine the sum of the consecutive odd numbers
from 1 to 79 by visualizing L-shaped figures forming a square.
Factors and Primes
6. All possible rectangular arrays that can be constructed with exactly 4 tiles, exactly 7 tiles,
and exactly 12 tiles are diagrammed below. Use the tiles from your Manipulative Kit or from
the Virtual Manipulatives to form all possible rectangular arrays that can be constructed for
the remaining numbers from 1 to 12 (remember a square is a rectangle). Sketch these arrays
in the spaces provided below.
Rectangles with 1 tile
Rectangles with 2 tiles
Rectangles with 3 tiles
Rectangles with 4 tiles
Rectangles with 5 tiles
Rectangles with 6 tiles
Rectangles with 8 tiles
Rectangles with 9 tiles
Rectangles with 11 tiles
Rectangles with 12 tiles
4
2
2
Rectangles with 7 tiles
7
Rectangles with 10 tiles
12
2EEttE8
6
3ffi±3
4
~
.
.
..--....
~~------------------
-....-
Activity Set 4.1
Models for Even Numbers, Odd Numbers, Factors, and Primes
101
7. The dimensions of a rectangular array are factors of the number the array represents. For
example, the number 12 has a 1 X 12 array, 2 X 6 array, and 3 X 4 array, and the factors of
12 are 1,2,3,4,6, and 12. Use your results from activity 6 to complete the following table by
listing all the factors of each number and the total number of factors. Then extend the table
to numbers through 20.
Number
Factors
Number
of Factors
1
2
3
4
1,2,4
3
1,7
2
5
6
7
8
9
10
11
12
1,2,3,4,6,
12
6
13
14
15
16
17
18
19
20
8. Numbers with more than one rectangle, such as 12 and 4, are called composite numbers.
Except for the number 1, all numbers that have exactly one rectangle are called prime
numbers.
a. Write a sentence or two to explain how you can describe prime and composite numbers
in terms of numbers offactors.
*b. How does the number 1 differ from the prime and composite numbers?
102
Chapter 4
Number Theory
c. Is there another counting number, like 1, that is neither prime nor composite? Explain
why or why not in terms of rectangles.
d.
Some numbers are called square numbers or perfect squares. Look at the sketches in
activity 6. Which numbers from 1 to 12 do you think could be called square numbers?
State your reason.
*e. Examine the rectangles representing numbers that have an odd number of factors. What
conclusions can you draw about numbers that have an odd number of factors?
9. The numbers listed in your chart have either 1, 2, 3, 4, 5, or 6 factors.
a. Identify another number that ha exactly 5 factors. Are there other numbers with exactly
5 factors? Explain.
b. Find two numbers greater than 20 that have more rectangles than the number 12. What
are the factors of each of these numbers?
10. The numbers 6, 10, 14, and 15 each have exactly 4 factors. Looking at the factor for these
numbers, determine what special characteristics the numbers with exactly 4 factors possess.
List a few more numbers with exactly 4 factors, and explain bow you could generate more
of the numbers.