MAT 156

Palya's Problem Solving Steps

1. Understand the Problem

2. Devise a Plan

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Some examples are to look for patterns, draw a diagram or symbols, act problem out, break it down, solve a simpler problem, or work backwards.

3. Impliment your plan

4. Looking back

Sets

Equal sets

Equivalent Sets

Subset

Set complement

Proper Subset

Set intersection

Set union

Set difference

Problem solving tool: Venn Diagram

Models/Context

Discrete (set) Model- Counted quantities

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Characterized by combining two sets of discrete objects. Ex: markers, fruits, animals

Continuous(number line) Model- Measured quantities

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Characterized by combining two continuous quantities. Ex: time, distance, area, volume

Properties of Addition

a

Closure Property of addition

Closure property of addition on whole numbers(W)

Commutative property of addition on whole numbers (w)

Associative property of whole numbers(w)

Identity property of addition of whole numbers (w)

Addition Tables

use the commutative property

the numbers are diagonal and appear in bands

Properties of Subtraction?

Closure property of subtraction on whole numbers(w)

No

Commutative property of subtraction on whole numbers(w)

No

Associative property of subtraction on whole numbers(w)

No

Identity property of subtraction on whole numbers(w)

No

Subtraction Algorithiums

Scratch Method

European Method

Properties of Multiplication

Closure property of multiplication on whole numbers (w)

Commutative Property of multiplication on whole numbers(w)

Associative property of multiplication on whole numbers(w)

Identity proerty of multiplicatin on whole numbers(w)

Zero property of multiplication on whole numbers(w)

Distributive property of multiplication over addition/subtraction on whole numbers (w)

Division Models/Context

Partition (equal Sharing)

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Need to know: Quantity starting with, and the number of groups.Will find: the size of each group

Measurement (repeated subtraction)

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Need to know: Quantity starting with, and the size of each group.Will Find: the number of groups

Number Theory

Even Numbers

The # is a multiple of 2, 2n

Odd numbers

if is one more than even #, 2n+1

Factoring

Prime Numbers

Composite Numbers

Greatest Common Factor

a

Use Cuisenaire Rods to visualize

Least Common Multiple

Use Cuisenaire Rods to visualize

Relatively Prime

Two numbers don't share any common factors

Modualar Clocks

a

Addition

move clock wise

Arithmetic Proerties

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Closure Property- if we add any two values on the clock do we get another value on the clock.Commutative- a+b(mod 12) = b+a(mod 12)Identity- a+?(mod 12)=aInverse- a+b(mod 12)=0

Subtraction

subtract the number by moving counter clockwise

Multiplication

repeated addition

Arithmetic Properties

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Closure Property- if we multiply any 2 values on the clock, do we get another value on the clock.Commutative- a x b(mod 12)= b x a(mod 12)Identity- a x ?(mod 12)=aInverse- a x b(mod 12)= 1

Division

Inverse multiplication

Decimals

addition

Line up decimal points to line up place value.

Can only add zero's to the left of the decimal.

Subtraction

Must subtract numbers in order given.

Line up decimal points

Multiplication

Doesn't matter which number is on top.

Don't have to line up decimals

Division

When you move the decimal point over on the divisor you have to do the same for the dividend.

After the decimal point the place value has th at the end.

Sequences

Arithmetic (common difference)

Rule: an=a1+d(n-1)

Geometric (common ratio)

Rule: an=a1 x r exponent n-1

Recurrence Relationship

current term is dependent on previous term

One-to-one correspondence

For each element of A there is an element of set B to match it with no extra elements and no repeated use of an element.

Ways of recording numbers

Tally System

Egyptian Symbols

Mayans (base 60)

Babylonians

Roman(Symbols in specific order)

Hindu-Arabic (usage of digits/numerals, and place values)

Base Blocks

Addition Algorithiums

Lattice Method

Scratch Method

Low Stress

Any column first

Left to Right Addition

Subtraction Models/Context

Take-away

Have initial quantity and remove a specified amount

Comparison

compare relative sizes of 2 quantities, determine how much larger or smaller one quantity is compared to the other.

Missing addend

Subtraction is inverse operation of Addition

Four Fact Families

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An example is:3 +7=107+ 3=1010 - 3=710 - 7=3

Has two addition and two subtraction facts that go together

Multiplication Models/context

Multiplcation as repeated addition of whole numbers (discrete)

Repeated addition Continuous (number Line)

Area Model

Cartesian Product Context (Can use tree diagram)

Multiplication Algorithiums

Partial Product

Lattice Method

Division is inverse operation of multiplication

Four fact Families

Has two multiplication and two division facts that go together

Divisibility

Equivalant Statements

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Assuming bIa then,b is a factor of a.a is a multiple of b.b is a divisor of a.a is divisible by b.

Tests

2

3

4

5

6

8

9

10

Algorithms

Array

Partial-Quotients

Scaffolding

Column

Fractions

Interpretations

Part-Whole

2/3 Represents 2 parts of a whole that was divided into 3 parts.

Division

Copies of a Unit Fraction

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Is the accompaniement to part-whole. An example is 2/3 is two copies of the unit fraction 1/3.

Ratio

Pattern Blocks

Help Student visualize fractions when working with shapes.

Egyptian Fractions

all must be unique unit fractions

Write as a sum of unit fractions

Improper/Mixed

Simplified

Add/Subtracting

Improper method

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Advantages: no regrouping of fraction needed, looks like part-whole context, process is similar to multiplication/division of fractions.Disadvantages: work with larger numbers, more opportunity for mistakes, multiply/divide within problem.

Mixed fraction form

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Advantages: no conversion, strengthens conception of place value, more consistent with add/sub on all types of numbers.Disadvantages: regrouping of fractions

Multiplication

Area Model

Division

Parition(Long Division)

Repeated Subtraction (Division of Fractions)

Need to know the size of the group and find the number of groups

Ratio

a

may look like fraction

Compares two quantities regardless of whether the units are same.

A quantitative relationship showing number of times one contains or is contained within another value.

Absolute Reasoning

Relative Reasoning

Proportion

A statement that 2 ratios are equal

Cross Multiplication

Shortcut to multiply both sides by LCD

Proportional Reasoning

Understand rational numbers

Ratio Sense

Unitizing

Reconizing quantities you have and how they change

Relative Thinking

Integers

a

Opposite of an integer

a and -a

number line approach

a,-a are both the same distance from zero

Chip method

Absolute Value

Number line approach

helps visualize

Chip method

Look for the magnitude of the number

Ordering integers

numbers to the left of -3 are less than<

numbers to the right of -3 are greater than >

Arithmetic with integers

Addition

use the discrete set model of addition

Subtraction

Take away model

Multiplication

Repeated addition

Division

Repeated subtraction model

Partition model

Percents

Per 100

Can use 100 grid to change fraction to percent.

Decimal Fraction

Whole number over a power of 10.

Terminating Decimal

A decimal that doesn't go on forever.