Math 365

Definition of Divisibility for Integers

If a/b is a unique integer, then a is divisible by b, or equivalently b divides a written as bla.

Divisibility Test for 2

If, and only if, the number is even.

Divisibility Test for 3

If, and only if, the sum of its digits is divisible by 3.

Divisibility Test for 11

If, and only if, the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11. (if the sums are different, subtract the smaller from the greater).

Divisibility Test for 9

If, and only if, the sum of the digits of the whole number is divisible by 9.

Divisibility Test for 8

If, and only if, the three rightmost digits of the whole number represent a number divisible by 8.

Divisibility Test for 4

If, and only if, the two rightmost digits of the whole number represent a number divisible by 4.

Divisibility Test for 6

If, and only if, the whole number is divisible by both 2 and 3.

Integers

Include all numbers: negative, zero, positive numbers

Whole Numbers

Include all positive numbers and zero

Natural Numbers

Include all positive numbers but not zero

Pattern Model for Multiplication of Integers

Multiply the same integer by consecutive numbers including positive, negative, and zero. A pattern will form.

Is zero defined as odd, even, or neither?

Neither

Is 1 Prime, Composite, or Neither?

Neither. There are not 2 distinct divisors.

Distributive Property of Multiplication over Subtraction for Integers

No parentheses at the end result. a(b-c)=a*b-a*c and (b-c)a=b*a-c*a.

Without division: Is 24,013 divisible by 12?

No. 12l24,000 but 12 does not divide 13. So, 12 does not divide 24,013.

Without division: Is 2*3*5*7+1 divisible by 6?

No. 6l2*3*5*7 or 6l210 but 6 does not divide 1. So, 6 does not divide 2*3*5*7+1.

Are the number of divisors of a perfect square even, odd, sometimes even and sometimes odd?

Odd. A number can only be counted once. 9: 1,3,9 4: 1,2,4 36: 1,2,3,4,6,9,12,18,36

Chip Model for Multiplication of Integers

Positive and negative integers are represented by different colored chips. Black for positive, red for negative. Ex: 3(-2) is represented by 3 groups of 2 red chips. There are 6 chips altogether, but since they're red, it is -6. Ex: -3(-2) is explained as "remove 3 groups of -2". So, start with the value of zero by having 3 groups of 2 in both black and red. By removing 3 groups of -2 (red), you are left with 3 groups of 2 (black) representing 6.

Chip Model for Subtraction of Integers

Positive and negative integers are represented by different colored chips. Start out with the first number (3), add the next number (-2) in both colors, remove the negative colored chips, and see how many are remaining (5).

Chip Model for Addition of Integers

Positive and negative integers represented by different colored chips. Use one-to-one correspondence and see how many of what color is remaining.

Number-Line Model for Addition of Integers

Shows which direction you are suppose to move according to the problem.

Charged-Field Model for Subtraction of Integers

Similar to Chip Model, but using + and - charges instead of colored chips.

Charged-Field Model for Multiplication of Integers

Similar to chip model but using + and - charges instead of chips

Charged Field Model for Addition of Integers

Similar to chip model but using + and - charges.

Are the number of divisors of a composite number even, odd, sometimes even and sometimes odd?

Sometimes odd and sometimes even. 9: 1,3,9; 12: 1,2,3,4,6,12

Pattern Model for Addition of Integers

Start with whole number addition then work your way to adding negative numbers to the same integer. This will display a pattern the student can see.

Subtraction using Adding the Opposite Approach

Subtracting an integer is the same as adding the opposite. Ex: 3-(-2)=3+2 (Leave, Change, Opposite)

Division word problem

There are six cookies and three kids. How many cookies does each kid get? or There are six cookies and each kid gets three cookies. How many kids get cookies?

True or False: If 8ln, then 2ln

True.

12/4

True: 12/4=3

4l12

True: 12=4*w

How to explain Division of Integers

Use Models of Multiplication

Associative Property of Multiplication of Integers

Yes. (a*b)*c=a*(b*c).

Identity Property of Multiplication of Integers

Yes. 1 is the unique integer such that for all integers a, 1*a=a=a*1.

Without division: Is 24,036 divisible by 12?

Yes. 12l24,000 and 12l36. So, 12l24,036.

Distributive Property of Multiplication of Integers

Yes. a(b+c)=a*b+a*c.

Closure Property of Multiplication of Integers

Yes. a*b is a unique integer.

Commutative Property of Multiplication of Integers

Yes. a*b=b*a.

Closure Property of Subtraction of Integers

Yes. a-b=c where c is a unique integer. The only property gained from whole numbers to integers.

"Divided by" symbol

a straight, diagonal line

"Divides" symbol

a straight, vertical line

Mission Factor for Division

a/b=c true, if and only if, c is the unique whole number such that b*c=a. 18/3=__ 3*__=18

Why are mental arithmetic and estimation important?

counting money shopping, grocery store cooking time distance etc.

How do you pronounce "-x"?

"opposite of x" Ex: x=7: -7, x=-6:6, x=0:0

Use Distributive Property of Multiplication over Subtraction for Integers to simplify (-3)(x-2).

(-3)(x-2)=(-3)(x)-(-3)(-2)=-3x-(-6)=-3x+6

Zero Property of Multiplication of Integers

0 is the unique integer such that for all integers a, a*0=0=0*a.

12-9/3*2+5=__

11. Use PEMDAS

Repeated Subtraction for Division

18/3=6 18-3=15-3=12-3=9-3=6-3=3-3=0

Odd Number

A whole number that leaves a remainder of 1 when divided by 2.

Even Number

A whole number that leaves a remainder of zero when divided by 2.

Why is 2 the only even Prime Number?

All other even numbers are divisible by 2.

Absolute Value

Always positive because you're counting the distance from zero, not focused on the actual value of the number itself.

Prime Numbers

Any positive integer with exactly 2 distinct, positive divisors.

Composite Numbers

Any whole number greater than 1 that has a positive factor other than 1 and itself.

Number-Line Model for Subtraction of Integers

Draw a number line on the board. Have student start at 0. If solving the expression: 3-(-2), have the student walk three steps in the positive direction representing the 3. When you subtract, turn around. When counting a negative number, walk backwards. So the student will turn around and walk backwards 2 times, representing the -2. They will end up at 5. So, 3-(-2)=5.

Number-Line Model for Multiplication of Integers

Draw a number line on the board. Have student start at zero. If solving expression 3*4, the student will jump 3 spots, 4 times, and will end up at 12. If solving the expression 3*-4, the student will jump backwards 3 spots, 4 times, and end up at -12. If solving -3*4, the student will turn around and jump 3 spots, 4 times, and end up at -12. If solving -3*-4, the student will turn around and jump backwards 3 spots, 4 times, and end up at 12.

The Fundamental Theorem of Arithmetic

Each composite number can be written as a product of primes in one, and only one, way except for the order of prime factors in the product.

Are the number of divisors of a prime number even, odd, sometimes odd and sometimes even?

Even. Only divisors of a prime number are 1 and itself.

True or False: If 2ln, then 8ln

False. 2l12 but 8l12

If 5l(a+b), then 5la and 5lb

False: 7+3=5 * 2 but, 7 does not equal 5*w and 3 does not equal 5*w.

0l0

False: 0 does not equal 0*w b\c "w" has to be a unique whole number. "w" could be any number in this situation.

12l4

False: 4 does not equal 12*w

Definition of Less Than for Integers

For all integers a and b, a is less than b, written a<b, if, and only if, there exists a positive integer k such that a+k=b.

Definition of Integer Division

For all integers a and b, a/b is the unique integer c, if it exists, such that a=b*c.

Equation

Has an equal sign

Expression

Has no equal sign

Prime Factorization

A factorization that contains only prime numbers. Ex: 12=3*2*2

0l3

False: 3 does not equal 0*w

How to get the Prime Factorization?

Repeated division: 360/2=180/2=90/2=45/3=15/3=5/5=1 Factor Tree: 360=6*60, 6=2*3 and 60=6*10, 6=2*3 and 10=2*5

Factorization

The expression of a number as a product of its factors. Ex: 12=3*4 or 12=2*6 or 12=1*12