Homeschool Math, www.HomeschoolMath.net website logo

Home | Blog | Amazon Store | Newsletter  

Ebooks | Worksheets | Curriculum Guide | Lessons & Articles | Reviews |
May newsletter     An online tutor helps you teach math at home!

The ideas in this lesson are taken from Multiplication Division 3 ebook. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.


Factoring, part 3
Free lesson plan from HomeschoolMath.net

To find all the prime numbers less than 100 we can use the sieve of Eratosthenes.

1 2 3 4 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
 51  52  53  54  55   56  57  58  59 60
 61  62  63  64  65  66  67  68  69 70
 71  72  73  74  75   76  77  78  79 80
 81  82  83  84  85  86  87  88  89 90
 91  92  93  94  95  96  97  98  99 100
     Cross out every other number starting at 2.
Cross out every third number starting at 3.
(You don't have to check every fourth number.  Why?)
Cross out every fifth number starting at 5
(You don't have to check every sixth number. Why?)
Cross out every seventh number starting at 7
(You don't have to check every eighth number. Why?)
(You don't have to check every ninth number. Why?)
(You don't have to check every tenth number. Why?)

The explanation why you don't have to check
numbers bigger than 10 is beyond this text.

The numbers that are not crossed out are primes.

List here all primes between 0 and 30:

____________________________________________________________ 
  

 

We have found how to know if a number is divisible by 2, 5, 10, 3, or 9.
We can use that in factoring.
260

/   \

26 × 10
     
260
/     \
26 × 10
/ \ / \
2 × __ × 2 × 5
     
  2
We know 260 is divisible by 10 and the division is easy to do. 26 is an even number, so it is divisible by 2. To find out the missing divisor, complete the  long division on the right.
135

/   \

5 × ?
2 7
1 3 5

0
3 5

5
0

  135

 /      \

5 ×

  27

|

 /   \

5 × 3

 ×  9

|

    /  \

5 × 3 × 3 × 3
We know 135 is divisible by 5 (why?).   Dividing 135 by 5 we find out that 135 = 5 × 27. Here is the complete factorization to prime numbers.

Example problems

1.  Factor the following numbers so the factors are prime numbers.  Do any needed long divisions in your notebook.

90
/ \

 

 

 

165
/ \

 

 

 

95
/ \

 

 

 

We can use divisibility rules in factoring bigger numbers, too.  Look at the examples.
 
5850
 /       \
10  × 585


5850
 /       \
  10  ×   585

  /   \        /  \

5 × 2 × 5 × ?
   
5850
 /       \
10   ×   585

   /  \         /    \

5 × 2 × 5  × 117
5850 ends in 0, so it is 
divisible by 10.
585 is divisible by 5
since it ends in 5.
10 we know is 5 × 2.

 

Dividing 585 by 5 we find
that 585 = 5  × 117.
5850
 /       \
10     ×   585

  /    \        /    \

5 × 2  × 5  × 117

 |      |      |       /  \

5 × 2 × 5  × 9 × ?
   
5850
 /       \
10    ×   585

    /    \       /   \

5 × 2 × 5  × 117

 /     /      /      /   \

5 × 2 × 5  × 9 × 13
   
5850
 /       \
  10   ×   585

  /   \        /  \

5 × 2 × 5  × 117

     /     /     /       /   \

5 × 2 × 5  × 9 × 13

  /     /     /      /  \       \

5 × 2 × 5  × 3 × 3 × 13
117 is divisible by 9
since the sum of its
digits is 1 + 1 + 7 = 9.
The division tells us
that 265 = 5 × 53.
 

 

2.  Circle or underline the number that the following numbers are divisible by.  Use divisibility rules.  Then pick the GREATEST divisor of those, do the division, and express the number as a product.  Do the long divisions in your notebook.  Follow the example given.

Number

Divisible by
(circle the numbers)

Division

Expression as
a product

260

10   5   3   2

260 10 = 26 260 = 10 × 26

96

10   5   3   2

96 __ = ___ 96  = ___ × __

95

10   5   3   2

95 __ = ___ 95 = ___ × __

132

10   5   3   2

132 __ = ___ 132 = ___ × __

3.  Use the previous exercise and factor these numbers.  Continue factoring to prime factors.

124

/       \

2  ×  

/          /      \

2  ×     × 

260
 /       \
10   ×     

/    \      /     \

96
/       \
  3   ×   

/          /    \

 

 

4.  Factor the following numbers in your notebook down to prime factors.  You can start factoring by any number that you notice it is divisible by!  Try first if the number is divisible by 10, then by 5, then by 2, then by 3 or 9.

306 =

990 =

945 =

 

Many times primes occur in pairs so that their difference is only two. For example, 5 and 7 are such a pair, so are 11 and 13.

Find all such pairs that are less than 100.  Use the sieve of Eratosthenes.
 How many such pairs are there?

continue to review

FREE math homework help on www.PhatMath.com
  • 40 Math forums for grades K-12 & College
  • Post/view math problems & solutions
  • Blogs for students, parents, teachers

Math Mammoth Grade 6 Worksheets Collection - not your run-of-the-mill worksheets!

Over 150 hand-crafted quality math worksheets covering all 6th grade topics.
See our samples!






Times Tales

Learn the upper
times tables with
mnemonic stories.

Learn more!




 

 

Learn...
- Multiply & divide 4-digit numbers
- Estimation
- Divisibility
- Factoring & primes

Self-teaching worktext - download today!

 

Teaching Tips and Math Lessons menu

All grades

Teaching math
Is your curriculum coherent?
How to motivate & prevent math anxiety
Keeping math skills sharp in the summer
Setting up equations for word problems
Interview with an astronomer
ADD/ADHD and diet
Omega-3 fats & intelligence

Elementary

Using calculator in elementary grades?
Teaching word problems

Middle

Balance illustrates equation solving
Why are fractions so difficult?
Negative or zero exponents
Number to zero power is one - 2 proofs
How to teach proportions
How to teach integers
Hands-on with pi
Sine in a right triangle
Rational numbers & converting repeating decimals into fractions
Why study math? square roots? algebra?
Square roots without a calculator?

High School

Why high school geometry is difficult
Help with high school geometry
What is proof?
Two-column proof vs. paragraph proof
Proving is a process - logarithm problem
Fascinating irrational numbers
Proof that square root of 2 is irrational
Rational numbers are countable
How does calculator find values of sine?
Fibonacci numbers and golden section
Why does the square root algorithm work?

 

 




HOMESCHOOL MATH NEWSLETTER

Homeschool Math Newsletter comes out once monthly, and includes math teaching articles and tips, Math Mammoth news, and all kinds of little "tidbits" such as math news, interesting links, or humor. The content is equally good for all of us who teach math (not just homeschooling parents).

When you subscribe, you will alsoreceive a GIFT of over 280 FREE math worksheets and sample pages from Math Mammoth books for grades 1-8.

Note: This is a double opt-in list. You will FIRST get an email that asks you to confirm your email address. PLEASE check also your SPAM/JUNK folder for this confirmation email.

Take a peek at the previous volumes here!.

The newsletter service is by Aweber.com. Your email address will not be shared, sold, or distributed to anyone else.

Google

Download 280+ QUALITY math worksheets
Link to us    Sitemap    Contact    About    Privacy    Advertise    Homeschool links    Educational links
Best Homeschool Sites on the Internet - Homeschool Top Sites
Copyright 2003-2008 Homeschool math
http://www.homeschoolmath.net/


 

 

P.S. ....

 

Check out these short video clips from my favorite band - Rockabilly.US. They play oldies, Elvis, country, and top hits of the 50s and 60s. It's a show with large band, costumes, choreography, little skits, "Kid Kritters", 6 dancing/singing girls, junior dancer troupe, etc.

"Be My Baby"
originally by the Ronettes

Impossible Dream (Elvis song)
Most importantly, the band wants to use the uplifting and energetic music to bring some fun, happiness and encouragement into people's lives. They're doing it selflessly to help others, and don't get paid for any of it themselves. And I admire that!

=> See more video clips at YouTube