welcome to math 6 today’s subject is… factors and prime factorization
TRANSCRIPT
Welcome to Math 6Today’s subject is…
Factors and Prime Factorization
“THE GREAT BACKYARD BIRD COUNT” is an annual 4-day event that engages bird watchers of all ages in counting birds to create a real-time snapshot of where the birds are.
Scientists and bird enthusiasts can learn a lot by knowing where the birds are. Bird populations are dynamic; they are constantly in flux.
No single scientist or team of scientists could hope to document and understand the complex distribution and movements of so many species in such a short time.
WHY COUNT BIRDS?
In 2012, GBBC participants submitted a record 104,151 checklists with 17.4 million individual bird observations!
Participants set new records in 22 states and in 6 Canadian provinces.
Across the continent and in Hawaii, participants identified 623 species!
The Eastern Bluebird is one of the most beautiful songbirds of North America.
How many of these were spotted in 2012?
Whither the Blue Jay?
Many people think of Blue Jays as present year-round in consistent numbers; but this is not the case.
Blue Jays are migratory, but their numbers also fluctuate from year to year due to cycles in wild food abundance (specifically acorns).
GBBC data- Number of Checklists in North Carolina reporting
a Blue Jay sighting:
Year Durham, NC Wilmington, NC
2012 87 702011 80 44
2010 107 58
In a previous course, you learned to find the greatest common factor of two or more numbers. Let’s see if you remember how to do it.
In 2012, 87 checklists from Durham, NC and 70 checklists from Wilmington, NC reported the Blue Jay.
What is the greatest common factor of 87 and 70?
Connector
Suggestions-
• For each number, make a list of all its factors.
• Remember, factors always occur in pairs.
These Divisibility Rules may help:
2 if the last digit is 0,2,4,6 or 8
3 if the sum of the digits is divisible by 3
4 if the last two digits form a number that is divisible by 4
5 if it has a 0 or 5 in the ones place
6 if it is divisible by both 2 and 3
9 if the sum of the digits is a number that is divisible by 9
• 87 is not divisible by 2, 4, 5, 6 or 9.
You can tell from the divisibility rules.
• 87 is divisible by 3, since its digits add up to 15 (a multiple of 3.)
• We know 90 ÷ 3 = 30, right?
Since 87 is 3 less than 90, 87 ÷ 3=29.
• 87 is not divisible by 7 or 8 (since 7x12=84 and 8 x 11=88).
The factors of 87 are: 1, 87, 3, 29,
• Obviously 7x10 =70.
• Tip: When you know one pair of factors, you can often find another pair by doubling one of them and taking half of the other.
For example, take 7x10=70. Since 2x7=14 and half of 10 is 5, so 14x5=70.
• The same strategy will take us from 1x70=70 to 2x35=70. Get it?
The factors of 70 are:
• 70 is not divisible by 3, 6 or 9.(Use the divisibility rules to find out why.)
• Since 70 is not a multiple of 4, so 4 is not a factor of 70.
• 70 is not divisible by 8, 11 or 12, since 7 x 12=84 and 8 x 11=88.
The factors of 70 are:
1, 70, 2, 35, 5, 14, 10 and 7.
The factors of 70 are:
The Common Factors of 87 and 70…
The greatest common factor (GCF) of 87 and 70 is 1.
In fact, 1 is the ONLY common factor.
87’s other factors are both prime numbers.
The factors of 87 are: 1, 87, 3, 29,
The factors of 70 are: 1, 70, 2, 35, 5, 14, 10 and 7.
Now that you’re warmed up, lets look at some objectives for this lesson.
Remember, every lesson has at least one objective. It is either a new skill that you can use or a new concept that you didn’t understand before the lesson.
OBJECTIVE FOR THIS LESSON:
Each student will be able to:
Use a factor tree to find the prime factorization of any number up to 100.
Key Vocabulary for this lesson
Prime Number - A whole number that is divisible by exactly two numbers: itself and 1.
or
A whole number greater than one that has exactly two factors, itself and 1.
Prime Factorization – The prime factorization of a number is the number written as the product of its prime factors.
Example- The factors of 32 are 1 x 32; 2 x 16 or 4 x 8;
If we continue to factor the factors of 32, we get: 2 x 2 x 2 x 2 x 2 or 25
Key Vocabulary for this lesson
Multiple – The product of any number and a whole number is a multiple
When we say that 25 is ‘divisible’ by 5, that means that 25 is a ‘multiple’ of 5.
Key Vocabulary for this lesson
Composite Number - Any number that is divisible by more than two numbers
or
A number, greater than one, that has more than two whole-number factors.
Key Vocabulary for this lesson
Let’s take another look at the factors of 70.
The factors of 70 are:
1, 70, 2, 35, 5, 14, 10, and 7.
You can also write a number as the product of its prime factors. We call this the prime factorization of the number.
We make a diagram called a Factor tree:
70
35 * 2
* 75
Choose any two factors to begin. Keep finding factors until each ‘branch’ ends at a prime number.
When you get to a factor that is a prime number, circle it. That ‘branch’ is finished.
The prime factorization of 70 is 2 * 5 * 7
Notice that it doesn’t matter which two factors you start with, you will always end up with the same prime factors: 257
70
14
27
5
70
10
25
7* *
**
Here is a factor tree for 87.
The number itself is written at the top of the tree.
It is not necessary to write 1 as a factor since 1 is a factor of every number.
87
3 29
87 has only 3 x 29, both prime numbers
Let’s review Prime Numbers.
Every number is divisible by 1 and by itself.
A prime number is divisible ONLY by itself and 1. A prime number has ONLY two factors: itself and 1.
Using a factor tree we can find the prime factors for any composite number.
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
Prime Numbers are ---------- in Gray
Take a look at the prime numbers and composite numbers from 1-50.
Guided Practice:
Write the prime factorization of each number.
1. 38
2. 76
3. 60
4. 132
Guided Practice:
38=219
19 2
381.,
∗
•
38 2
76
219
Guided Practice:
∗
Guided Practice:
3.
60=2235 30 2
60
215
35
∗
∗
∗
Guided Practice:
4.
132=22311
66 2
132
116
32
∗
∗
∗
Tip: Your factor trees do not have to look exactly the same as my examples.
The key is that the set of
Prime Factors is the same.
Independent Practice:
1. Which is the prime factorization of 48?
a) 2 3
b) 24 3
c) 23 3
d) 2 34
*
*
*
*
What number does each of the other choices represent?
(Pause the slide show while you complete these problems.)
Independent Practice:
2. Which is the prime factorization of 150?a) 2 3 5b) 2 3 52
c) 22 3 5d) 23 55
*
**
***
*
What number does each of the other choices represent?
(Pause the slide show while you complete these problems.)
Independent Practice:
3. Write the prime factorization of each number:
a) 160
b) 92
c) 56
d) 132 (Pause the slide show while you complete these problems.)
Independent Practice Answers:
1. Which is the prime factorization of 48?
a) 2 3= 6
b) 24 3= 2x2x2x2x3= 48
c) 23 3= 2x2x2x3=24
d) 2 34= 2x3x3x3x3=162
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*
*
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Independent Practice Answers:
2. Which is the prime factorization of 150?a) 2 3 5= 30b) 2 3 52= 2x3x5x5=150c) 22 3 5= 2x2x3x5=60d) 23 55= 2x2x2x5x5=200
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Independent Practice Answers
160 2 * 80 2 * 40 2 * 20 2 * 10 2 * 5
3a) 25 *5=160
3b) 92 46 * 2 23 * 2
22*23=92
Independent Practice Answers
3c) 56 28 * 2 14 * 27 * 2
23*7=56
Independent Practice Answers
3d) 132 66 * 2 33 * 211*3
22*3*11=132
Independent Practice Answers
If you got those correct, you have met the lesson objective. Congratulations!
If you missed any, don’t worry. You’ll have plenty more opportunities to practice.
By the way, bigger numbers are not any harder to do. They just take longer.
GBBC data: Total numbers of bird seen:
YearEastern Bluebird
Blue Jay
2012 5,529 4,260
2011 4,684 3,758
2010 5,295 5,138
2009 4,424 4,616
2008 4,367 3,658
Challenge:
• In 2012, 4260 Blue Jays were spotted during the Great Backyard Bird Count. Can you find the prime factorization of that number?
• Solution on the next slide.
4260
2*2130
2*1065
3*355
5*71
Challenge- Solution:The prime factorization of 4260
is 22*3*5*71.
That’s a lot of Blue Jays!
• A factor tree is a diagram you can use to find the prime factors of any number.
• Every number has its own unique set of
prime factors.• Prime factorization means to express a
number using only its prime factors• Multiplication facts and divisibility rules
can help you to find prime factorization.
Conclusion:
Assignments-
1. Explain the difference between factors of a number and prime factors of a number.
Eastern Bluebird
Assignments-
2. The prime factorization of 50 is 252. Without dividing or using a diagram, find the prime factorization of 100.
3. Tell how you know when you have found the prime factorization of a number.
Assignments-4- Complete the attached assignment.
Painted Bunting
THAT’S ALL FOR TODAY. HOPE YOU ENJOYED HEARING MORE ABOUT MY INTEREST IN BIRDS!
PLEASE COMPLETE ALL THE ASSIGNMENTS BEFORE BEGINNING THE NEXT LESSON.