MAT 156 Frymire

Intergers and the Operations of Addition and Subtraction

Addition

Chip Model for addition- Black and red chips to represent positive and negative numbers.
Charge Field Model- Similar to chips but with + and - signs
Number Line Model- Always start at 0. Move left or right depending on the sign
Absolute Value- |x| and |-x| = x, -|x|= -x

Properties of Integer Addition
- Closure Property
- Commutative Property
-Associative Property
- Identity eelement of addition of integers- 0

Subtraction

Chip Model
Charge Field Model for Subtraction- same as addition
Number line for Subtraction- Same as addition, except, when you come across a negative number you TURN AROUND
Subtraction using the Adding the Opposite Approach- Example: 2-8= 2+8= -6 (keep the sign of the larger number)

Properties
- Cannot do commtative property nor associative with subtraction

Multiplication of Integers

Methods

- Pattern Model for Multiplication of Integers- 3(-2)=-6; 2 (-2)= -4... etc
- Chip Model and Charge Field Model- remember to use groups, not just the each chip (-3)(-2)> 3 groups are taken from 2 groups
- Number Line Model- Remember to use groups for multiplication

Properties of Multiplication of Integers

-Closure Property
- Commutative Property
- Associative Property of Multiplication
- Distributive Property of Multiplication over Addition of Integers
- Zero Multiplication Property of Integers
- Additive Inverse

Order of Operations on Integers

PEMDAS- This is the same as the order of operations for anything

Divisibility

Divisibility Rules:

2: If it ends with an even number- 165418
3: If the sum of the digits is divisible by 3- 969
4: If the last two digits are divisible by 4- 424
5: If it ends in 0 or 5- 65
6: If the sum of the digits are divisible by 2 and 3: 126
8: If the last 3 digits are divisible by 8- 1848
9: If the sum of the digits are divisible by 9- 126
10: If the number ends in 0

Prime and Composite Numbers

Prime Factorization

-Factor Tree
- Ladder Model- see diagram in notes
- Number of Divisors- group them together
-Sieve- the game

Prime number:
A number that has only two factors, 1 and itself

Composite number:

Numer in which there are more than two factors

Greatest Common Divisor/ Least Common Multiple

Greatest Common Factor: The greatest factor that can go into both or all numbers

Methods to Finding the GCF:
-Colored Rods Method
- The Interestection of Set Methods- List all of the factors and choose the greatest number that intersects
-Prime Factorization- Factor tree
Ladder Method- See diagram in the notes

Subtopic

Problem Solving

George Polya's 4 Step Problem Sovling

1. Understand the Problem

Look for patterns

2. Devising a plan

3. Carrying out the plan

4. Looking back (Check!)

Work Backwards

Patterns

Inductive Reasoning

Conjecture

Counter example

Arithmetic Sequence

Addition or subtraction

an= a1+(n-1)d

Fibonacci Sequence

No fixed difference

1,1,2,3,5,8,13, 21,34,55...

Add the last two sums

Geometic Sequence

Multiplication

an=a1 * r(n-1)< exponent, r= ratio

Numeration Systems

Hindu- Arabic Numeration System

All have base ten

2345= 2000+300+40+5

Tally Numeration System

Tallies- grouped in sets of 5

Roman Numerals

I=1
V= 5
X=10
L=50
C=100
D=500
M=1000

If you place a smaller number before a larger number, it means to subtract it. i.e. IV= 5-1=4

Base 5

1030five= (1*5^3) + (0*5^2) + (3*5^1) +(0*5^0)

Sets

One-to- One Correspondence

There is exactly one match per set.

{1,2,3} {a,b,c}

Equal Sets

The same numbers and amount of numbers in each set. Order does not matter

{1,2,3} {3,2,1}

Equivalent Sets

Must have the same number of items per set.

{a,b,c,d} {l,m,n,o}

Subsets

Contained within the other set

{1,2,3,4,5,6} {2,3,4}

Set Intercection

Venn Diagram

cUs

Set Union

The whole Venn Diagram

Set Difference

The compliment

Properties of Set Operations

Associative

Order is important. Also called grouping

Communative

Order doesn't matter

Cartesian Products

A X B (A cross B)

Addition and Subtraction

a+b=c
a+b> Addends
c> sum

"Mastering Addition"

Counting On

Doubles

Making 10 (and then add any leftovers)

Counting Back

Fact Families

Properties of Additions

Identity Property- a+0=a

Commutative Property- a+b=b+a

Associative Property- (a+b)+c= a+(b+c)

Closure Property- If a and b are whole numbers, then a + b is a whole number.

"Mastering Subtraction"

Inverse Operations

Take-Away Model

Missing Addend Model

Comparison Model

Number Line Model

Properties of Subtraction

Closure- {1,3,5,7,…} (3-5=-2) NO- Answer is not a WHOLE number

Associative (a-b)-c=a-(b-c) YES

Commutative- a-b=b-a NO

Identity- a-0=a YES However: 0-a=0 is not true.

Algorithms

Addition Algorithms

To help students understand algorithms, we should start with manipulatives. Children can touch, move around, and be led to developing their own algorithms.

b. After working with manipulatives, then move to paper/pencil operations.

c. Regroup or trade problems are then used to describe carrying.

Lattice Algorithm for Addition
Example: 3 5 6 7
+ 5 6 7 8
0/1/1/1
/8/1/3/5
9 2 4 5

Subtraction Algorithms

Use base-ten blocks to provide a concrete model for subtraction as we did in addition.
b. The concept of remove or take away is used.
c. Then paper/pencil algorithms are introduced.

III. Equal-Addition Algorithm
a. Based on the fact that the difference between two numbers does not change if we add the same amount to both numbers.
Example: 255 > 255 + 7 > 262 > 262 + 30 > 292
- 163 163 + 7 -170 -(170 + 30) - 200
92

Multiplication and Division

Multiplication

• Repeated-Addition Model
• We can use addition to put equal groups of numbers together to use multiplication. 3 + 3 + 3 + 3 = 12 (four groups of 3’s)
• Can be shown by number lines and arrays. (See pg. 143)
• The constant feature (+) on a calculator can help relate multiplication to addition. Example: + 3 = = = = 12

The Array and Area Model

• Cartesian-Product Model
Use of a tree diagram to solve multiplication problems (See pg. 146)
*Be aware of how multiplication is modeled:
*A X B, A(B), A B where A and B are the factors and A X B is the product

Properities of Multiplication

A. Closure property of multiplication of whole numbers- The set of whole numbers is closed under multiplication. That is, if we multiply any two whole numbers, the result is a unique whole number.
B. Commutative property of multiplication of whole numbers- For whole numbers a and b, a X b = b X a.
C. Associative property of multiplication of whole numbers- For whole number a, b, and c, (a X b) X c = a X (b X c)
D. Identity property of multiplication of whole numbers- There is a unique whole number 1 such that for any whole number a, a X 1 = A = 1 X a
E. Zero multiplication property of whole numbers- For any whole number a, a X 0 = 0 = 0 X a
F. Distributive property of multiplication over addition and subtraction- For any whole numbers a, b, and c, a(b+c)= ab + ac and a(b-c) = ab – ac
Example of how distributive property works:
7 X 13 = 7 X (10 + 3) = (7 X 10) + (7 X 3) = 70 + 21 + 91

Division

Set (Partition) Model- Set up a model of the total number of items in the problem then partition them into sets. Example: 18 cookies divided by 3 would be 3 sets of 6 cookies

b. Missing-Factor Model- Using multiplication, the number of groups times the unknown variable is equal to the total. Example: 3 X c = 18 By using multiplication, we know that 3 X 6 equals 18, thus c = 6.

Repeated subtraction model-
Example: 18 divided by 6 could be shown as 18- 6 = 12 – 6 = 6 – 6 = 0
or 18 - 6 -6 -6 = 0

V. Relating Multiplication and Division as Inverse Operations
*Division is the inverse of multiplication.
*Division with a remainder of 0 and multiplication are related.
*Note—is division closed, commutative, associative, and/or identity property?

VI. Division by 0 or 1 ( bottom of pg. 154 and see School Book Page on pg. 155)
• n divided by 0 is undefined (there is no answer to the equivalent multiplication problem.)
• 0 divided by n = 0
• 0 divided by 0 is undefined also.

Order of Operations

Parenthesis
Exponents
Multiplication/ Division
Addition/ Subtration

Functions

A FUNCTION from set A to Set B is a correspondence from A to B in which each element of A is paired with one, and only one, element of B

Functions as Equations

Functions as Arrow Diagrams

Functions as Tables and Ordered Pairs

Functions as Graphs
Horizontal- inputs
Vertical- outputs

Relations
A relation from Set A to set B is a correspondence between elements of A and element of B, but unlike functions, do not require that each element of A be paired with one, and only one, element of B.

• Every function is a relation, but not every relation is a function

The Set of Rational Numbers

Rational numbers are all numbers

Can be proper or improper fractions

Denseness of Rational Numbers:

Find two numbers between two different fractions by multiplying denominators

Addition, Subtraction, and Estimation with Rational Numbers

Be able to model fractions with pictures

Use a number line to represent a fraction and estimate. Remember to always start at 0.

Additive Inverse Property: a/b, the additive inverse would be -a/b. Additive inverse is the opposite of the original number. Should add up to 0.

Addition Property of Equality: Two equivilant fractions added to the same fraction will equal eachother.

Use a number line to estimate fractions. Be careful with - numbers though- remember the closer to zero on the left side is the greater number.

Multiplication and Division of Rational numbers

Multiplication can be modeled with repeated addition.

"Of the" in a word problem means multiplication.

Multiplicative Identity: The number 1 is unique- when multiplied by a number, it is the number.

Multiplicative Inverse: The opposite. When multiplied will equal 1.

Division: Keep, Switch, Switch

a^m * a^n= a^m+n

a^-n= 1/a^n

a^m/a^n= a^m-n

(a^m)^n= a^mn

Introduction and Operations on Decimals

Need to know the ones, tens, and hundreds places in decimals.

Terminating decimals are ones that end. Order them with the greatest one, then tens, then hundreds and so on.

Repeating Decimals: Ones that don't end and repeat the same numbers to infinity. Order them in the same manor as ordering terminating decimals.