# Factors for the First 50 or 100 Numbers

## Planning notes and worksheet

### Introduction

Pose a question about the number of factors numbers have and invite students to discover a relationship. Have students make collections of factors, organize them in a list or table. Use the table as a Bridge to discover relationships. The relationship of factors to their product and common factors to the different numbers they can make. Move students toward and understanding of multiplicative relationship for factors instead of additive. See worksheets below and completed - factors to 50 table and total number of factors from 1-10 for numbers 1-50 ; factors to 100 table and total number of factors from 1-20 for numbers 1-100

### Facts, Concepts, and Generalizations

Number value and operations

• Factors are numbers that are multiplied together and result in a product. a * b = c; a and b are factors and c is the product.
• A prime number has only two factors - itself and one. One is not a prime, it has only one factor.
• Every number has a unique (one and only one) prime factorization (Fundamental Theorem of Arithmetic).
• Primes only have two factors so if you multiply it by another prime, that makes three factors (2*1) [2 primes] (3*1) [2 primes or one more - since 1 is in both].
• When multiplying primes the exponent is one less than the number of factors ( 32 is 9 which has three factors)
• Multiplicative (2n) is different than additive (n+n). Skip counting is additive (0) (2) (2 + 2) (2 + 2 + 2) . When students count groups and then announce the product you can conclude they are unitizing and using a multiplicative strategy.
• Multiply a number by another number and the result will have as many factors as the product (multiplicative relationship) of the factors (4 has 3 factors; 3 has 2 factors; 12 has 6 factors; 3 factors * 2 factors = 6 factors).

Data analysis

• Ordering by value can help discover relationships
• Putting data in a chart can help distinguish relationships
• Ordering by equality can help distinguish relationships

Reasoning and proof

• Finding a counter example will disprove a conjecture.
• Disproving a conjecture can provide insight.
• When explaining (justify) my ideas it is important to select and include accurate information that is necessary and sufficient.
• It's helpful to see and hear other peoples' ideas to get different views and insights.
• Systematically exploring all possibilities can be used as a proof.

### Mathematical focus questions

• What is a factor?
• How many factors does a number have?
• What number form 1-50 has the most factors?
• What number form 1-50 has the least number of factors?
• How many factors for the number one? Two? ...
• Is there a relationship for the number of factors?
• Can we figure it out?
• What happens when the numbers are doubled?
• What if we sort numbers (1-50) by the number of factors for each number?
• How can we use the relationships we found to fine more factors? or Fill in the table?
• What number form 1-100 has the most factors?
• What number form 1-100 has the least number of factors?

Worksheet 1

Number Factors Number of Factors
1 1*1 ---------> 
2 1*2 ---------> [1, 2]
3 1*3 ---------> [1, 3]
4 1*4, 2*2 ---------> [1, 2, 4]
5 1*5 ---------> [1, 5]
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

Worksheet -2

# Numbers that Have 1-10 Factors or First 50 Numbers Sorted By Their # of Factors

One
Factor
***
Two
Factors

Three
Factors
√√√
Four
Factors

Five Factors
∆∆∆
Six
Factors

Seven
Factors
ΩΩΩ
Eight
Factors

Nine
Factors
∑∑∑
Ten
Factors

Dr. Robert Sweetland's Notes ©