NCDPI – AIG Instructional Resource: Background Information
Resource Title: Narcissistic Numbers
Time Frame: 20-30 minute class
Subject Area/Grade Level (s): Math/6
Common Core/Essential Standard Addressed:
Apply and extend previous understanding of arithmetic to algebraic expressions.
6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
Common Core Standards for Mathematics
1. Make sense of problems and persevere in solving them.
5. Use appropriate tools strategically.
6. Attend to precision.
Additional Standards Addressed: NA
Brief Description of Lesson/Task/Activity: This lesson is to be used when students are working with evaluating expressions using whole
number exponents.
Type of Differentiation for AIGs (include all that apply):
Adaptations for AIGs:
x Content
x Process
x Enrichment
x Extension
Acceleration
Product
Explanation of How Resource is Appropriate for AIGs: On a basic level, all students should be able to evaluate expressions using exponents of
whole numbers. This lesson goes beyond the general curriculum by introducing narcissistic numbers which is an extension topic. It requires that
students understand how to find a narcissistic number and apply their knowledge to any given number.
Needed Resources/Materials: NA
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Sources:
•
Wikipedia provides a description of narcissistic numbers (also called Armstrong numbers and plus perfect numbers).
•
http://en.wikipedia.org/wiki/Narcissistic_numbers
•
The On-line Encyclopedia of Integer Sequences has a list of several narcissistic numbers.
•
http://oeis.org/A005188
TEACHER NOTES: NA
NCDPI AIG Curriculum Resource Outline
STAGE ONE: ENGAGE
Students should be able to readily find the value of a single digit raised to a given power. A calculator will make this activity much less
cumbersome and eliminate multiplication errors. A possible warm-up for this activity could be:
Solve the following:
42 =
83 =
74 =
Solutions:
42 = 16
83 = 512
74 = 2401
STAGE TWO: ELABORATE
Have students find the following sums. Ask students if they notice something about one or more of the sums compared to the numbers used in
the sums.
1+5+3=
12 + 5 2 + 3 2 =
13 +23 +33 =
14 + 2 4 + 3 4 + 4 4 =
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
4+0+7=
42 + 0 2 + 7 2 =
43 + 0 3 + 7 3 =
44 + 0 4 + 7 4 =
1+6+3+4=
11 + 61+ 31+ 41 =
12 + 6 2 + 3 2 + 4 2 =
13 + 6 3 + 3 3 + 4 3 =
14 + 6 4 + 3 4 + 4 4 =
Answers:
1+5+3=9
12 + 52 + 32 = 1 + 25 + 9 = 35
13 +53 +33 = 1 + 125 + 27 = 153
14 + 54 + 34 = 1 + 625 + 81 = 707
4 + 0 + 7 = 11
42 + 02 + 72 = 16 + 0 + 49 = 65
43 + 03 + 73 = 64 + 0 + 343 = 407
44 + 04 + 74 = 256 + 0 + 2401= 2657
1 + 6 + 3 + 4 = 14
12 + 62 + 32 + 42 = 1 + 36 + 9 + 16 = 62
13 + 63 + 33 + 43 = 1 + 216 + 27 + 64 = 308
14 + 64 + 34 + 44 = 1 + 1296 + 81 + 256 = 1634
Narcissistic numbers are numbers that equal the sum of each of its digits raised to the power of how many digits there are in the original
number. If there are 3 digits, then we need to find the sum of each digit raised to the third power. If there are 4 digits, we need to find the sum
of each digit raised to the 4th power and so on. If the sum found is equal to the original number, it is narcissistic.
Is 153 narcissistic? (from above)
There are 3 digits in the number; therefore, we need to find the sum of each digit raised to the third power.
13 + 53 + 33 = 1 + 125 + 27 = 153
Is 1234 narcissistic?
There are 4 digits in the number; therefore, we need to find the sum of each digit raised to the fourth power.
14 + 24 + 34 + 44 = 1 + 16 + 81 + 256 = 354
The sum is not equal to 1234, so it is not narcissistic.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Ask students why they think these numbers are called narcissistic. Students may be unfamiliar with the term narcissistic (vain) and could benefit
from explanation of the word before relating it to the numbers.
Students may solve the following individually, in pairs, or in small groups.
Which of the following numbers are narcissistic?
370, 407, 535, 1634, 2416, 44332, 54748
Solution:
Narcissistic: 370, 407, 1634, 54748
Not Narcissistic: 535, 2416, 44332
STAGE THREE: EVALUATE
Students can complete an exit slip with the following question:
Is the number 9474 narcissistic?
Solution – Yes it is.
94 + 44 + 74 + 44 = 6561 + 256 + 2401 + 256 = 9474
TEACHER NOTES: All singles digits 1-9 are narcissistic as any number in this range raised to the first power is itself. Zero is not a narcissistic number.
There are no 2 digit narcissistic numbers.
Depending on timing of class, the teacher may decrease the amount of numbers the students are asked to investigate.
Teachers can also have students connect these numbers to Narcissus in mythology.
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project