Illuminations: Factor Trail Game

# Factor Trail Game

 When students play the Factor Trail game, they have to identify the factors of a number to earn points. Built into this game is cooperative learning — students check one another's work before points are awarded. The score sheet used for this game provides a built-in assessment tool that teachers can use to check their students' understanding.

### Learning Objectives

 Students will: Practice identifying the integer factors of numbers up to 100.

### Materials

 Factor Trail Game Calculators (optional)

### Questions for Students

 Which number on the game board has the most factors? [The numbers 60, 72, 90 and 96 all have 12 factors.] For which number on the trail will a player earn the most points? [The sum of the factors of 96 is 252, offering the most points of any number on the board.] In general, how many points are earned for a prime number? [For a prime number p, then p + 1 points are earned. That is, the number of points is one more than the number itself, since the only factors of a prime number are the number and 1. For instance, 18 points are earned for the prime number 17, whereas 41 points are earned for the prime number 41.]

### Assessment Options

 Collect the score sheets of all students. The score sheets can be used to determine if students were correctly finding the factors of numbers. One of the benefits of using the score sheet occurs when a 0 is entered in the "Points Earned" column. This indicates that the student made a mistake when finding the factors of that number, so it is easier to identify areas of difficulty.

### Extensions

 Change the numbers on the game board. Note that all of the numbers are less than 100. For a more advanced game, include numbers in the hundreds or thousands. Students can use their score sheets as the beginning of a list of abundant, deficient, and perfect numbers. A number is said to be abundant if the sum of its proper divisors is greater than the number itself. A number is said to be deficient if the sum of its proper divisors is less than the number itself. A number is said to be perfect if the sum of its proper divisors is equal to the number itself. All numbers can be categorized as either abundant, deficient, or perfect. The mathematician Leonhard Euler identified the sigma function σ(n), which gives the sum of the positive divisors of n. Using this notation, then abundant numbers are those with σ(n) > 2n, deficient numbers are those with σ(n) < 2n, and perfect numbers are those with σ(n) = 2n. Students can use the Abundant, Deficient, Perfect activity sheet to keep track of the numbers 1–100. Students can also research the fascinating life of Euler. One of the topics that Euler studied was amicable pairs (an amicable pair consists of two integers for which the sum of proper divisors of one number equals the other number, and vice versa). Because amicable pairs also depend on the factor sum of numbers, they are closely related to abundant, deficient, and perfect numbers.

### Teacher Reflection

 Did students enjoy the game? While students played the game, were they able to focus on the mathematics of the activities? What common errors did students make while finding the factors of a number? What misconceptions might have led to those errors, and how might those misconceptions be corrected?

### NCTM Standards and Expectations

 Number & Operations 3-5Develop fluency in adding, subtracting, multiplying, and dividing whole numbers; Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.
 This lesson prepared by G. Patrick Vennebush.

1 period

### Lessons

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.