MAT 156 Frymire

Computing for Humans

por Dénes Harmath

Ontario Chemistry Curriculum

por Mitchell Smith

NISSAN MOTOR CORPORATION

por Iván Ortega

RISK MANAGEMENT

por Mindomo Team

Chapters 6 & 7

Introduction and Operations on Decimals

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

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

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

Multiplication and Division of Rational numbers

(a^m)^n= a^mn

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

a^-n= 1/a^n

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

Division: Keep, Switch, Switch

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

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

"Of the" in a word problem means multiplication.

Multiplication can be modeled with repeated addition.

Addition, Subtraction, and Estimation with Rational Numbers

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.

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

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.

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

Be able to model fractions with pictures

The Set of Rational Numbers

Denseness of Rational Numbers: Find two numbers between two different fractions by multiplying denominators

Can be proper or improper fractions

Rational numbers are all numbers

Chapters 3 & 4

Functions

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

Functions as Graphs Horizontal- inputs Vertical- outputs

Functions as Tables and Ordered Pairs

Functions as Arrow Diagrams

Functions as Equations

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

Multiplication and Division

Division

Order of Operations

Parenthesis Exponents Multiplication/ Division Addition/ Subtration

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.

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?

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

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.

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

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

• 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

The Array and Area Model

• 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

Algorithms

Subtraction Algorithms

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

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.

Addition Algorithms

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

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

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

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

Addition and Subtraction

Properties of Subtraction

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

Commutative- a-b=b-a NO

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

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

"Mastering Subtraction"

Inverse Operations

Number Line Model

Comparison Model

Missing Addend Model

Take-Away Model

Properties of Additions

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

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

Commutative Property- a+b=b+a

Identity Property- a+0=a

"Mastering Addition"

Fact Families

Counting Back

Making 10 (and then add any leftovers)

Doubles

Counting On

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

Chapters 1 & 2

Sets

Cartesian Products

A X B (A cross B)

Properties of Set Operations

Communative

Order doesn't matter

Associative

Order is important. Also called grouping

Set Difference

The compliment

Set Union

The whole Venn Diagram

Set Intercection

Venn Diagram

cUs

Subsets

Contained within the other set

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

Equivalent Sets

Must have the same number of items per set.

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

Equal Sets

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

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

One-to- One Correspondence

There is exactly one match per set.

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

Numeration Systems

Base 5

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

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

Tally Numeration System

Tallies- grouped in sets of 5

Hindu- Arabic Numeration System

All have base ten

2345= 2000+300+40+5

Patterns

Geometic Sequence

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

Multiplication

Fibonacci Sequence

No fixed difference

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

Add the last two sums

Arithmetic Sequence

an= a1+(n-1)d

Addition or subtraction

Inductive Reasoning

Counter example

Conjecture

Problem Solving

George Polya's 4 Step Problem Sovling

4. Looking back (Check!)

Work Backwards

3. Carrying out the plan

2. Devising a plan

1. Understand the Problem

Look for patterns

Chapter 5

Greatest Common Divisor/ Least Common Multiple

Subtopic

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

Prime and Composite Numbers

Composite number: Numer in which there are more than two factors

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

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

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

Multiplication of Integers

Order of Operations on Integers PEMDAS- This is the same as the order of operations for anything

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

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

Intergers and the Operations of Addition and Subtraction

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

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