# MAT156 - Mathematics/Elementary Teachers I

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MAT156 - Mathematics/Elementary Teachers I

## 1. Test 1

### 1.1. Chapter 1 and 2

1.1.1. Chapter 1

1.1.1.1. Chapter 1-1 Mathematics and Problem Solving

1.1.1.1.1. George Polya's Four-Step Problem Solving Process

1.1.1.2. Chapter 1-2 Exploration with Patterns

1.1.2. Chapter 2

1.1.2.1. Chapter 2-1 Numeration Systems

1.1.2.1.1. 1. Hindu-Arabic Numeration System

1.1.3. Chapter 1

1.1.3.1. 1-1 Mathematics and Problem Solving

1.1.3.2. 1-2 Explorations with Patterns

1.1.4. Chapter 2

1.1.4.1. 2-1 Numeration Systems

1.1.4.1.1. 1. Hindu-Arabic Numeration System

## 2. Test 3

### 2.1. Chapter 5

2.1.1.1. A. chip model B Charged - field model C. Number line model. D. Pattern model. E. Absolute Value.

2.1.2.1. a. Closure property of addition of integers

2.1.2.2. b. Commutative property of addition of integers

2.1.2.3. C. Associative property of addition of integers

2.1.2.4. d.. Identity element of addition of integers 0

2.1.2.4.1. 0 is the unique integer such that for all integers a, 0

2.1.2.4.2. 0+a=a=a+0

2.1.3. Uniqueness of the Additive Inverse

2.1.3.1. For every integer a, there exists a unique -a , the additive inverse of a, such that a+-a=0 =-a+a Also, -(-a)=a and-a+-b=-(a+b)

2.1.4. Integer Subtraction

2.1.4.1. Properties of Subtraction: Cannot do commutative nor associative with subtraction.

2.1.4.2. subtraction using the missing addend approach 5-3=n if and only if 5=3+n

2.1.5. Multiplication and Division of Integers

2.1.5.1. Multiply or divide a positive and positive the answer is positive.

2.1.5.2. Multiply or divide a positive and a negative the answer is negative.

2.1.6. 5-3 Divisibility

2.1.6.1. Divisibility Rules

2.1.6.1.1. for 2 if its (last ) units digits is divisible by 2

2.1.6.1.2. for 3 if the sum of its digits is divisible by 3 for example 123= 1+2+3=6 and 6 is divisible by 3

2.1.6.1.3. for 4 if the last two digits represent a number divisible by 4 for example 248 -48is divisible by 4

2.1.6.1.4. for 5if the last digit is 0 or 5

2.1.6.1.5. for 6 if the integer is divisible by 2 and 3

2.1.6.1.6. for 8 if the last three digits of the integer represnt a number divisible by 8

2.1.6.1.7. for 9 if the sum of the digits of the integer is divisible by 9

2.1.6.1.8. for 10 if the last digit is zero

2.1.6.1.9. for 11 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 odd powers of 10 is divisible by 11

2.1.7. 5-4 Prime and composite number

2.1.7.1. number 1 is not a prime or composite number

2.1.7.2. prime numbers

2.1.7.2.1. there are only two factors . for example 2: (2 X 1)

2.1.7.3. Composite Numbers

2.1.7.3.1. There are more than two factors for example 4 : ( 4x1 & 2x2)

2.1.7.4. Prime Factorization

2.1.7.4.1. 1- composite numbers can be expressed as products of two or more whole numbers greater than 1 for example, 18=2x9, 3x6,2x3x3 each expression is a factoriztion.

2.1.7.4.2. 2- prime factorization - a factorization containing only prime numbers.

2.1.7.4.3. 3- Factor tree

2.1.7.4.5. 5- Number of Divisors

2.1.8. 5-5 Greates common Divisor and Least common multiple

2.1.8.1. 1- Greatest Common Divisor - GCF or GCD

2.1.8.2. 2- Least common Multiple (LCM)

## 3. Test 4

### 3.1. chapter 6

3.1.1. Rational numbers

3.1.1.2. Uses of Rational number

3.1.1.2.1. 1. Division problems 3x= 1 is 1/3 2. part of whole 3. Ratio 4. probability

3.1.2. improper fraction

3.1.3. proper fraction

3.1.3.2. proper fraction : the numerator is smaller than denominator

3.1.4. session 6-3 multipication and division of Rational numbers

### 3.3. Chapter 7-1 & 7-2 Introduction and operations on Decimals

3.3.2. Rounding to the nearst ten https://www.youtube.com/watch?v=yqDsbO1G0kc

## 5. Chapter 3

### 5.1. Addition of Whole Numbers

5.1.1. Set Model

5.1.1.1. 1 Number Line (Measurement) Model

5.1.1.1.1. 2 Ordering Whole Numbers

5.1.1.1.2. always start at zero with your first number

5.1.2.1. A. Counting On

5.1.2.2. B. Doubles

5.1.2.3. C. Making 10 (and then add any leftovers) exmple: 8+5=(8+2)+3=13

5.1.2.4. D. Counting Back

### 5.2. Subtraction of Whole Numbers

5.2.1. Subtraction is the inverse of addition.

5.2.2. A. Take Away Model - You have 8 blocks, take away 3 blocks = 8 - 3= 5 blocks left.

5.2.3. B. Missing Added Model - Relates addition and subtraction 3 +___= 8

5.2.4. C. Comparison Model - Juan's 8 blocks, Susan's 3 blocks

5.2.5. D. Number Line Model

5.2.6. E. Property of Subtraction - Which addition properties work for subtraction

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

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

5.2.6.3. c. Commutative - a-b=b-a NO

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

### 5.3. Algorithms - A systematic procedure used to accomplish an operation.

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

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

5.3.1.3. C. Regroup or trade problems are then used to describe carrying.

5.3.1.4. D. Lattice for Algorithm for Addition

5.3.2. II. Subtraction Algorithms

5.3.2.1. A. Use base-ten blocks to provide a concrete model for subtraction as we did in addition.

5.3.2.2. B. The concept of remove or take away is used.

5.3.2.3. C. Then paper/pencil algorithms are introduced.

5.3.3.1. A. Based on the fact that the difference between two numbers does not change if we did add the same amount to both numbers.

5.3.4. IV. Understanding Addition and Subtraction in Bases Other Then Ten

### 5.4. Session 10 - (3-3) Multiplication and Division of Whole Numbers

5.4.1. I. Multiplication of Whole Numbers

5.4.1.2. The Array and Area Model

5.4.1.3. Cartesian-Product Model

5.4.2. II. Properties of Whole Number Multiplication

5.4.2.1. A. Closure property of multiplication of whole numbers

5.4.2.2. B. Commutative property of multiplication of whole numbers

5.4.2.3. C. Associative property of multiplication of whole numbers

5.4.2.4. D. Identity property of multiplication of whole numbers

5.4.2.5. E. Zero multiplication property of whole numbers

5.4.2.6. F. Distributive property of multiplication over addition and subtraction

### 5.5. Algorithms for whole-number Multiplication and division

5.5.1. 1- multiplication algorithms

5.5.1.1. a) single digit number times two digit number

5.5.1.1.1. b) two digit numbers times two digit

5.5.2. Division algorithms

5.5.2.1. A. the typical instruction for long division is Divide multiply subtract check bring down