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May 2013

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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 and prime numbers
Free lesson plan from HomeschoolMath.net

This 'lesson' should actually be divided into several lessons for the student.

In any multiplication the numbers that are multiplied are called factors and the result is called a product.

factor		factor		product
7	×	6	=	42

From this multiplication fact we can make two division facts:
42 ÷ 6 = 7 and 42 ÷ 7 = 6.
Therefore, 42 is divisible by both 6 and 7. We say 6 and 7 are divisors of 42.

Example problems

1. Express each number as a product of two factors. Sometimes there are several ways of doing this and you can choose which way you like, but don't use 1 × the number itself. Look at the example. This process is called factoring.

product

factors

product

factors

product

factors

10 =

50 =

24 =

5 × 2

12 =

22 =

110 =

94 =

121 =

40 =

2. Express each number as a product of two factors and then make division facts and state by which numbers it is divisible. Look at the examples. Sometimes there are many ways that you can express a number as a product of two factors.

Products:		Division facts:
10 = 5 × 2 10 = 10 × 1		10 ÷ 5 = 2 10 ÷ 2 = 5 10 ÷ 10 = 1 10 ÷ 1 = 10

10 is divisible by 1,2, 5, and 10.

Products:

Division facts:

20 = 5 × 4

20 = 10 × 2

20 = 20 × 1

20 ÷ 5 = 4
20 ÷ 4 = 5

20 ÷ 10 = 2
20 ÷ 2 = 10

20 ÷ 20 = 1
20 ÷ 1 = 20

20 is divisible by 1, 2, 4, 5, 10, and 20.

Products:		Division facts:
15 = __ × __ 15 = __ × 1

15 is divisible by ________________.

Products:		Division facts:
14 = __ × __ 14 = __ × 1

14 is divisible by ________________.

Products:		Division facts:
11 = __ × 1

11 is divisible by ________________.

Products:

Division facts:

18 = __ × __

18 = __ × 1

18 is divisible by ________________.

3. Use the previous exercise as a help and make a divisibility list of numbers between 1 and 20. Circle every number that is divisible only by 1 and the number itself. Such numbers are called prime numbers. (Usually 1 is excluded and is not counted as a prime number.)

number	divisible by: (divisors)	number	divisible by: (divisors)	number	divisible by: (divisors)	number	divisible by: (divisors)
1	1	6		11		16
2		7		12		17
5		10	1, 2, 5, 10	15		20

factor		factor		product
1	×	11	=	11

Some numbers only have two divisors: 1 and the number itself. Such numbers are called prime numbers. 11 is one of them.

Earlier we found that prime numbers less than 20 are

2, 3, 5, 7, 11, 13, 17, and 19.

1 is usually not counted as a prime number.

product		24
		/ \
factors	4	×	6

	24
/ \
4	×	6
/ \		/ \
2 × 2	×	2 × 3

Here 24 is factored to 4 × 6. We can continue by factoring 4 and 6.

We cannot factor 2 or 3 any further (except to 2 = 2 × 1 and 3 = 3 × 1) because they are prime numbers. So 24 = 2 × 2 × 2 × 3.

/ \

5 × 6

/ / \

5 × 2 × 3

We cannot factor 5 any further because it is a prime number. So 30 = 5 × 2 × 3.

/ \

3 × 7

Both 3 and 7 are prime numbers, so we cannot factor any further. So 21 = 3 × 7.

Example problems

1. Fill the table and circle each prime number between 20 and 30.

number	divisible by: (divisors)	number	divisible by: (divisors)
21		26
22		27
23		28
24		29
25		30

2. Factor the following numbers so the factors are prime numbers.

18
/ \

6
/ \

14
/ \

25
/ \

33
/ \

15
/ \

Prime numbers are like building blocks of all numbers. They are the first and foremost, and other numbers are 'built' from them. ALL numbers can be factored down so the factors are just prime numbers. That is sort of amazing - think about it!
For example, try these bigger numbers:

	100
/ \
10 × 10
/ \ / \
	×

56
/ \

64
/ \

84
/ \

Does it matter how you start your factorization? Try and see.
Does it make any difference if you start out by factoring 24 into 6 × 4, 8 × 3 or 12 × 2?

	24
/ \
6	×	4
/ \		/ \

	24
/ \
8	×	3
/ \		\|

	24
/ \
12	×	2
/ \		\|

We can do the factoring process backwards, too. Let's start with the building blocks and build numbers:

2 × 5	×	2 × 2
\ / \ /
10	×	4
\ /
	40

2 × 3	×	2 × 3	×	2
\ /		\ /		\|
6	×		×	2
\|		\ /
	×
\ /
72

2 × 7	×	2	×	3
\ /		\ /
	×
\ /

3. Build numbers from primes.

2 × 5 × 11
\ / |

3 × 2 × 2 × 2
\ / \ /

2 × 3 × 7
\ / |

Continue to Sieve of Eratosthenes

Math Lessons menu

Place Value Ideas Teaching tens and ones Tens and ones practice Counting in groups of ten Skip-counting (0-100) Comparing (0-100) Cents and dimes Place Value - hundreds Comparing (0-1000) Place value - thousands Comparing thousands Rounding Place value - big numbers	Add/Subtract lessons Missing addend concept (0-10) Addition facts with 6 Addition & subtraction connection Add/subtract whole tens (0-100) Easy principle of adding ones Fact families for basic facts Sum over next whole ten Add 2-digit numbers mentally Carrying to tens Rounding to nearest ten Carry two times Regrouping or borrowing in subtraction Subtract 2-digit numbers mentally	Multiplication & Division Multiplication Multiplication concept as repeated addition Multiplication on number line Commutative Multiply by zero Word problems Order of operations How to help students with multiplication tables Drilling tables of 2, 3, 5, or 10 Drilling tables of 4, 11, 9 Multiplying by whole tens Multiplying by 100 Distributive property Multiplying in columns the easy way Multiplying in columns Estimation when multiplying Basic division Division as making groups Division/multiplication connection Division is repeated subtraction Zero and one in division Division that is not exact (remainder) Divisibility Long division How to teach long division Long division as repeated subtraction Why long division works Remainder & long division Two-digit divisor Divisibility Divisibility within 0-1000 Divisibility rules Factoring Sieve of Eratosthenes
Fraction Lessons Understanding fractions Part of whole group Mixed numbers Mixed number to fraction Fraction to mixed number Adding like fractions Equivalent fractions Adding unlike fractions Adding mixed numbers Subtracting mixed numbers Subtracting mixed numbers 2 Part of whole group 2 Measuring in inches Comparing fractions Simplifying fractions Whole number times a fraction Fraction times a fraction Multiplication as an area Simplify before multiplying Division of fractions - explain why it works Divide fractions by whole number Dividing fractions by fractions Dividing mixed numbers	Geometry Lessons Lines, rays, and angles Measuring angles Parallel & perpendicular Triangles Angles in a triangle Equilateral & isosceles triangles Circles Symmetry Polygons Perimeter Area of rectangles Area of right triangles Area of parallelograms Area of triangles Area versus Perimeter Angles in Polygons (PDF) Review: Area of Polygons (PDF) Surface Area (PDF)	Decimals Lessons Tenths Tenths and place value Add decimals with tenths Multiply decimals with tenths Decimals: hundredths Hundredths and place value Thousandths Add decimals with hundredths Add decimals with hundredths 2 Multiply with hundredths Money problems. Comparing decimals

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The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.

Factoring and prime numbers Free lesson plan from HomeschoolMath.net

Example problems

Example problems

Factoring and prime numbers
Free lesson plan from HomeschoolMath.net