CHAPTER

Activity Set 4.1

VirtualManipulatives

..• crB.J.J• I • • •

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 andexplored by students. For example, building rectangular arrays with a set of tile can stimulatequestions about divisibility and prime, composite, square, even, and odd numbers .... 1

MODELS FOR EVEN NUMBERS, ODD NUMBERS,FACTORS, AND PRIMES

PURPOSE

To use models to provide visual images of basic concepts of number theory .

www.mhhe.comlbbn

MATERIALSColor Tiles from the Manipulative Kit or from the Virtual Manipulatives.

INTRODUCTION

Some problems in number theory are simple enough for children to understand yet are unsolv-able by mathematicians. Maybe that is why this branch of mathematics bas intrigued so manypeople, novices and professionals alike, for over 2000 years. For example, is it true that everyeven number greater than 2 can be expressed as the sum of two prime numbers? It is true for thefirst few even numbers:

4=2+2 6=3+3 8=3+5 14 = 7 + 710 = 5 + 5 12 = 5 + 7

However, mathematicians bave not been able to prove it is true for all even numbers greaterthan 2.

The ideas of odd, even, factors, and primes are basic concepts of number theory. In thisactivity set, these ideas will be given geometric form to show that visual images can be associatedwith them.

ICurriclllllm and Evaluation Standards for School Mathematics (Reston, VA: National Council of Teachers ofMathematics, 1989): 9 J -93.

95

Activity Set 4.1 Models for Even Numbers, Odd Numbers, Factors, and Primes

Even and Odd Numbers1. 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 oftile figures, Extend the sequence by drawing the figures for the numbers 6, 7, and 8.

2 3 4 5 6 7 8

Describe, in your own words, how the figures for the even counting numbers differ fromthose for the odd counting numbers in the sequence above.

2. Here is a tile sequence for the first four even numbers.

tst 2d 3d 7th4th

2 6 84

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 isthe 125th even number?

125th even number _

c. Describe in words what the figure for the nth even number would look like. Write amathematical expression for the nth even number.

nth even number _

97

98 Chapter 4 Number Theory

3. This tile sequence represents the first four odd numbers.

1st 2d 3d 4th 5th 8th

3 5 7

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 the15th odd number?

15th odd number _

c. Determine the 50th, lOOth, and nth odd numbers by imagining what their figures wouldlook like, based on the model shown above.

100th odd number _50th odd number _ nth odd number _

*d. When the odd numbers, beginning with 1, are arranged in consecutive order, 7 is in thefourth position and 11 is in the sixth position. In what position is 79? 117? (Hint: Thinkabout the tile figures for these numbers. You may wish to draw a rough sketch of thefigures.)

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 dif-ferences below by drawing similar diagrams.

----- -

Odd number Even + Odd number = Odd numberEven number

a. The sum of any two even numbers

*b. The sum of any two odd numbers

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 rearrangedas follows:

D~~~~

R"""9'dI;"< D ~ ~ LOriginal sequence

Repositioned tiles

a. The L-shaped figures for the first five consecutive odd numbers can be pushed togetherto form a square. What does this tell you about the sum of the first five consecutive oddnumbers?

100 Chapter 4 Number Theory

*b. Visualize the L-shaped figures for the first 10 consecutive odd numbers. What sizesquare can be formed from these figures? What is the sum of the first 10 consecutiveodd 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 numbersfrom 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 fromthe Virtual Manipulatives to form all possible rectangular arrays that can be constructed forthe remaining numbers from 1 to 12 (remember a square is a rectangle). Sketch these arraysin 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

4

2

2

Rectangles with 7 tiles Rectangles with 8 tiles Rectangles with 9 tiles

7

Rectangles with 10 tiles Rectangles with 11 tiles Rectangles with 12 tiles

12

2EEttE86

3ffi±34

..--.... -....-~ ~~------------------. .

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. Forexample, the number 12 has a 1 X 12 array, 2 X 6 array, and 3 X 4 array, and the factors of12 are 1,2,3,4,6, and 12. Use your results from activity 6 to complete the following table bylisting all the factors of each number and the total number of factors. Then extend the tableto numbers through 20.

Number Factors Number of Factors

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1,2,4

1,7

1,2,3,4,6, 12

3

2

6

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 primenumbers.

a. Write a sentence or two to explain how you can describe prime and composite numbersin 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? Explainwhy or why not in terms of rectangles.

d. Some numbers are called square numbers or perfect squares. Look at the sketches inactivity 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. Whatconclusions 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 exactly5 factors? Explain.

b. Find two numbers greater than 20 that have more rectangles than the number 12. Whatare 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 thesenumbers, 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 moreof the numbers.