Math Reflection

Math Reflection
Part 1– Effective Teaching of Mathematics After taking this course I have personally learned so much about mathematics and effective ways on
how to teach certain topics in my future classroom. I believe that this course was taught in a way for me to advocate for how I teach myself
mathematics. When I knew I was taking this class I was very nervous at first. Mathematics is not my strongest subject so automatically I felt as
though this class was going to be a long struggle for me. Once I had a couple class periods I grew more and more comfortable with the math we
were learning. I learned to teach math to myself again. I had to teach myself math concepts that I learned way back in elementary school that I
completely forgot how to do. In this class we learned a lot about how elementary math is done now a days. Also how now a days there is more than
just using the traditional algorithm to solve equations.
I have learned math along with my students at my placement. At my placement I observed my students using the basic operations with distributive
property, partial quotients with division and much more. One of the really cool things I learned from my placement students is the box method which
is done with multiplication. I at first thought it would be like lattice but it's not. For example say if you had the numbers 56 and 6, you would draw a
rectangle box and split it in the middle. Then you would write 50 in the first column on top and then 6 next to it. The number that goes on the other side
is 6. So then you would multiply 50 and 6 to get 300 in the first box. After you would do 6 times 6 with the other box which is 36. Once you have those
two numbers you add them together to get the answer. I was never taught to solve multiplication that way so that was something with mathematics that
I learned along with my students. I do believe that thoughtful listening is one of the most effective qualities of a good teacher. I practiced this
thoughtful listening when I listened to other people participate and show their work in class. I also practiced this thoughtful listening when I taught my
unit lesson on math word problems. I took the time while teaching my lessons to listen to every student explain how they got
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Essay About My Math Teaching Experience
My math teaching experience went with students who are in third grade. In my group I had four students, where was one ELL student. They were
struggling in math as per their teacher, especially in subtracting big numbers with zeroes. I had to reteach subtraction, three or two digit number, as
needed, from three digit number. And concentrate on subtractions from zeroes. Everything started great, the kids were exiting, as of they are going
out of the room. We sat down in the hall and it was not comfortable.The kids were sitting near the wall, I was sitting in front of them in the middle
of the walking area. As of kids were sitting on the floor and me too, we were several times distracted by other kids or adults, who was walking
through the hall, it was always disturbing us. I started my lesson with math solving problem and my students easy were able to figure out what
operation that needs to do. They all said subtraction. I said great. Then, I asked them to solve this problem independently. Student 1, was just sitting,
Student 2 used her fingers to count, Student 3 did very fast, but not correctly and Student 4 just copied that incorrect answer. Then, I asked to share
their answers. Student 3 raised her hand and said, it was easy, we have 0–9, will be 9. Then 0–5, will be 5. Then 4–2 will be 2. So, the answer is 259.
From this answer I understood, that my students understand that ones must be under one, and tens under tens, hundred under hundred, but students

Disadvantages Of Multiplication In Mathematics
Based on the Council of Teacher of Mathematics,(1993) "Multiplication involves the counting of units of a size other than one." The repeated addition
definition while is a useful link between multiplication and addition is limiting if it is students' only concept of multiplication.
The meaning of the multiplication sign, "Г—", depends on the language of the speaker. In Japanese it always means "multiplied by." "3 Г— 4" and "3
times 4" mean, "Three multiplied by four," or four groups of three items. In English, however, the sign means either "times" or "multiplied by,"
where"3 times 4," denotes three groups of four items, and "three multiplied by four" means four groups of three items. Therefore, in English the sign
"Г—" has two interpretations ... Show more content on Helpwriting.net ...
Basic knowledge on numbers and Mathematical skills.
4.Pedagogical readiness is the "Students" understanding of the materials they use as they learn Mathematics.
5.Maturation readiness is the "Students" level of mental maturity (each person passes through four stages of mental maturity).
The use of games and concrete materials can aid in pupils' recognition of the importance of Mathematics and its many different real applications.
Educators views on the use of concrete materials and Manipulatives in the classroom
Manipulative materials are objects that pupils can feel, touch, handle and move. The National Council of Teacher of Mathematics( 1993), proposes that
pupils and teachers often view the use of manipulatives"as play time", but stress that using manipulatives is any excellent way to help learners make
faster connection between mathematical ideas. It is also established that learning is enhanced when pupils are exposed to concepts in varying
manipulative context. Learners who are at the concrete level deal with manipulative materials as they discover solution to problems. Many if not all
games, involve object manipulation at some

Numbers In Norse Mythology
Numbers have had a fascinating presence throughout history. There is evidence of the use of numbers dating back at least twenty–thousand years, such
as the discovery of the Ishango bone. The Chinese, Mayans, Romans, Babylonians, and Greeks all had numbering systems.They were all unique and
they have all played a role in the use of Numbers in their everyday lives. Numbers have also played a significant role in magic and divination.The
I–Ching system from the Chinese, the tarot, and numerology are all some very popular forms. A Greek philosopher named Pythagoras is credited for
modern day numerology. Arithmancy is another form of divination based on assigning numbers to a word or phrase.This can be seen in the Hebrew
Kabbalah.
Numbers are all ... Show more content on Helpwriting.net ...
Odin, hung himself in Yggdrasil( the tree of life), he spent nine days and nine nights hanging in the tree. He made this sacrifice of himself to gain
power over the runes and learn of all their secrets. During Ragnarok Thor fights the sea serpent Jormungand, after killing the serpent, Thor takes his
final nine steps and then dies from all the poison the serpent spit on him. In Norse mythology there are nine worlds, Asgard– home of the gods called
Aesir, Alfheim– the land of light elves, Vanaheim– the home of the Vanir gods, Midgard– the human world, Jotunheim– the land where the Giants
dwelled, Svartalfheim– where the dark elves lived, Nidavellir– dwarf territory, Muspelheim– the land of fire that Surt ruled, Niflheim– The mist world.
In the center of Niflheim was the world of the dead, ruled by Hel. Yggdrasil (the tree of life) is in the midst of these nine worlds.These are just a few of
the ways the number nine is significant in Norse mythology. The number nine is in the Christian Bible. According to bible, number 9 has got a lot of
spiritual significance. When Jesus Christ was nailed on the cross, he dies in the ninth hour. Also Jesus appears a total of nine times to his apostles and
disciplines after his resurrection. As specified by Saint Paul, there are nine spiritual gifts of God which are wisdom, knowledge, faith, gift of healing, to
operate miracles, prophecy, discreetness of spirits, tongues and to interpret them. Also Saint Paul listed nine fruits of spirit which are love, joy. Peace,
patience, kindness, goodness, truthfulness, gentleness and self control. The number is is seen throughout different stories in the Christian Bible and is
considered a sacred number. In Chinese culture they like the number nine alot. The number nine to the means eternal and everlasting. On Valentine's
day a man usually gives a woman either 99 or 999 red roses to symbolize their everlasting love. In traditional Chinese

Place Value
The ability to make sense of numbers is necessary in order to solve a wide range of mathematical problems, both in further education and every day
life. A teachers understanding of place value and the base–ten system is paramount for developing a student's ability to confidently approach number
operations. To underpin and guide a student's knowledge of place value, clear insight into what a student knows about the concept, is pertinent to the
continuation of learning and building of understanding.
Rationale
Educators must draw upon their professional judgement to make astute teaching and learning decisions, to extend a student's learning of place value
and the base–ten system. Effective teachers will; implement well–planned mathematical activities to help engage learning, use observation and focused
question to see where support is needed, and provide opportunities for growth and understanding of place value (Department Of Education Western
Australia [DOEWA], 2013). One–on–one interviews are an invaluable diagnostic assessment strategy that unravels, not only the ... Show more content
on Helpwriting.net ...
The essence of understanding place value, is not only knowing the values of the numbers depending on their position, but being able to rename their
values in order to problem solve (NSW Government, 2015). For example, the number 345 could be written as 34 tens and 5 ones, 345 ones or 3450
tenths (National Council for Curriculum and Assessment [2015], 2015). The most significant aspect is being able to comprehend that whilst there may
be different ways to write the number, it still has the same value. The Australian Curriculum recognises this importance for more complex learning in;
algebra, fractions, decimals and multiplicative thinking (Commonwealth Of Australian, 2009). Without this knowledge, problem solving becomes
limited, resulting in confusion and

The Development Of Place Value
1.3 The Development of Place Value
Place value is essential to developing number sense and without it students would not be able to give meaning to numbers. Place value underpins
important mathematical concepts, such as part–part–whole knowledge, estimations, mental strategies, flexible partitioning, and knowledge of
multi–digit operations (Dawson, 2013; Hurst & Hurrell 2014). Frequent hands–on counting experiences with concrete materials, models, resources and
activities are mandatory to progress students understanding of place value (Appendix A). Ross (2002, p. 420) states when students are provided with
conceptual problem–solving activities rather than procedural activities, a greater understanding of place value is developed. Moreover, multiple
embodiment experiences allow students to work flexibly with numbers. Research shows students' ability to work flexibly with numbers in different
contexts are limited if they are not provided with "...perceptually different models" (Reys et al., 2012, pg. 28). Hence, students' development of place
value relies heavily on conceptual learning with explicit multi–embodiment experiences and tasks using concrete resources. However, before students
can develop base–ten number and place value systems, educators must introduce pre–number concepts including classifications, patterns, conservation,
comparisons and one to one correspondence, group recognition (subitising), and counting strategies (Rey et al., 2012). These concepts are

HOW CHILDREN UTILIZE THEIR MATHEMATICAL MIND AS PART OF...
"Dr Maria Montessori took this idea that the human has a mathematical mind from a French philosopher Pascal and developed a revolutionary math
learning material for children as young as 3 years old. Her mathematical materials allow the children to begin their mathematical journey from a
concrete concept to abstract idea". With reference to the above statement please discuss how these children utilize their mathematical mind as part of
their natural progression, to reason, to calculate and estimate with these Montessori mathematical materials in conjunction with their aims and
presentations? What is a mathematical mind? The Mathematical Mind' refers to the unique tendencies of the human mind. The French philosopher
Blaise Pascal said... Show more content on Helpwriting.net ...
A significant discovery that Dr. Montessori made was the importance of offering indirect preparation for the math materials while children were in the
sensitive periods for movement and the refinement of the senses. It is through children's work with the Exercises of Practical Life and Sensorial
materials that they first encounter and experience the concepts of measurement, sequence, exactness, and calculation Sensorial education is the basis
of mathematics. Dr. Montessori said that children are sensorial learners. They learn and experience the world through their five senses. So sensorial
education helps the child to create a mental order of the concepts he grasps using his five senses. "The skill of man's hand is bound up with the
development of his mind, and in the light of history we see it connected with the development of civilization." – Maria Montessori, THE ABSORBENT
MIND, Chap 14. pg. 138 Montessori firmly believed that the 'hands' are the mother of skills. By providing Montessori sensorial materials to the child
she was convinced that correct manipulation with quality and quantity would certainly create a lasting impression in the child's mind with the
understanding of mathematics. We place materials quite intentionally on trays, we color code activities, materials are displayed

Lesson Reflection Paper
At the beginning of the Kindergarten class the class did their normal routine of calendar time. The teacher tells them to go to the carpet for calendar
time and that gets them on task. They count the number of days they have been in school out–loud. They also count the days with straws and put them
in a ones, tens, or hundreds pocket. The students also keep record of the number of days they have been in school with change. They count out
change to the number of days they have been in school. They say how much each coin is worth and who is on it. Lastly, they add a gumball to their
jar of gumballs and say how many they have in total. This is a normal everyday morning routine that involves math. TMPs 3 and 6 are used for this
routine and SMPs 4, 6, and 8 are used during this time. This activity takes around thirty minutes everyday. I think doing this everyday has really
benefited the students. A student found a dime outside and he brought it to me. He knew it was a dime, how much it was worth, and even the
president that is on the dime SMP 6. I observed a forty–five minute second grade lesson using the Invagations program. The lesson was called how
many stickers how much money (cents). The lesson began with the students coming to the carpet and reviewing the value of coins. After that the
students were shown stickers in a base ten blocks format. They were told to come up with an answer of the total number of sticks and write it on their
white boards SMP 7. Then they were

Observation Is The Major Role Of The Teacher
Observation: Observation is the major role of the teacher. The teacher observes for routines and procedure to be observed. The teacher observes for
ground rule in the classroom and in individual materials. The ground rules are opportunity for the child to develop his/her will. The teacher observes
for pattern of order in the behavior of the child. The teacher observes for respect child to child, child to material, child to adult interaction.
Control of Error: is the built in aspect of the material also sometime found as coding on the back of the material.
For Example, in spindle box the extra or not enough spindle left is control of error. The material like matching numeral tonumber rods, teacher is the
control of error. Prepare of the environment: A prepared environment is an important role of Montessori classroom .the goal of prepared environment
is to make children safe and comfortable. A calm and orderly space enables the children to work on various activities of their choice at a pace they are
comfortable with. They also experience a mix of freedom and self–discipline in space that is designed to meet their development needs.
A teacher observes the child working with the materials and make changes to the environment. In mathematic materials, extensions and additional
problems are introduced to deepen child's understanding of materials. For example, variations introduced when child is ready with basic concepts of a
particular material to support child needs.
Relation

Place Value And Decision-Making Model
Introduction This report is about place value and how it should be approached based on the interview style diagnostic observations of year six student
'Ariel' and her knowledge and understanding of place value, including her ability to correctly use it with mathematical operations. The aim was to
diagnose areas that required learning and offered improvement opportunities. The aim was to then support this with the development of a teaching
plan. This was achieved by instructional activities through a series of tutoring sessions that worked towards refining her areas of knowledge and
improvement, and then extend on those ideas. After completing the diagnostic test, the areas Ariel struggled most with were multiplicative thinking,
partitioning,... Show more content on Helpwriting.net ...
They are selected to support and enhance the lesson objective (Reys et al., 2012, p. 43). There are ungrouped materials that can be formed in to
groups, for example, single beads or straws or anything that can be bundled into groups of say,10, or another amount. There are also pre–grouped
materials that are already bundled to represent an amount. The use of 'manipulative' materials can be useful visual aids and assist children grasp
concepts with a hands on approach, making it easier for a teacher to observe what a child is doing rather than thinking (Marshall & Swan,

Gnfgj
Perhaps the earliest device for working out sums was the abacus. This began as a clay tablet into which grooves were cut. Pebbles were then placed or
taken away from grooves to perform addition and subtraction. (Our word calculation comes from the Latin word calculus meaning 'pebble').
Because the pebbles were likely to become mislaid, they were later replaced by beads threaded on to wires and mounted in a frame. By moving the
beads backwards and forwards, addition, subtraction, division and multiplication could be done. In 1614 John Napier, an astronomer, invented a
ready–reckoned, known as Napier's bones, to help him make complex calculations accurately. From this was developed in 1621 the earliest form of the
slide rule.
The first ... Show more content on Helpwriting.net ...
These circuits are mainly used for solving a wide variety of complex engineering problems, like investigation of stresses in aircraft, ships and large
engineering structures. They can also be used to simulate and set up models of complex installations and study the effect of various operational factors
on the complex installations.
They can for example, be used to simulate the behavior of an aircraft in response to the actions of crew members. Using analog computers, apparent
equipment failures or other emergencies can be introduced for proper training of the crew. The first automatic analogcomputer designed to solve
complex differential equations, was described in detail in 1876 by the English Scientist William Thomson.
Digital Computers: These are used in commerce and industry for extensive arithmetical calculations which would otherwise require enormous clerical
effort. Such computers carry out mathematical operations with the variables expressed in the computer as numbers, usually in the binary system (given
below).
These numbers are recorded in the computer electronically, as a series of temporary magnets, each magnetized in one of the two possible directions.
The two magnetization direction corresponds two numbers of the binary system.
The first electronic digital computer, known as the Electronic Numerical Integrator and Computer or NIIAC was developed at the University of
Pennsylvania in 1947 by

2123 Base 3
Adding numbers in base 3 is quite simple. First of all you identify the problem, for this occasion the problem used will be 2123 +1223 knowing
that we will be adding in base three means we are restricted to only using the numbers 0,1,2. Now we are going to start by adding 2 and 2 together
which makes four, but you can't write down four so we must now subtract 3 so that it can give us a number to work with, and we are also
subtracting 3 because it's the base we are working with. Then, you must carry the 1 and write down a 1 down at the bottom too. Secondly, we are
going to now add the middle column, which is 1 and 2 also don't forget to add the 1 that we carried earlier.Four would be the answer but we can't
write down 4 so as previously done before we have to subtract 3 and carry a 1 and also write down 1 at the bottom of the problem because that is
the answer. Finally we move to the last column which is 2 plus 1 and the other 1 we previously carried. The answer to this is four again so you just
repeat the step we took earlier. Since we carried the one from the left over four we must now bring it down to our answer. At the end you should've
ended up with the number 1,111 if... Show more content on Helpwriting.net ...
First of all we have to find out what number stands for which letter in the alphabet. For this example we are going to be using A=01 and B=001 so
for example C would have to equal 0001 and D would be 00001 and so on. Since we now have our rules for what numbers mean which letters we can
move on to writing secret messages. With our given code we can now decipher any code, so then what would 0010100001 mean? It is easier to
break down the code first. since all of our numbers end in 1 we can start there and example of this would be the following: 001,01,00001. Now that
we have our code separated the final step is to look for what the numbers stand for in this case 01=A, 001=B and 00001=D so this must mean that
BAD is our secret

Math Observation
In the beginning of my observations the students in the classroom had been practicing their addition and subtraction in a one digit numbers. However,
as the year went on my cooperating teacher started to get her students to be solving two–digit addition and subtractions problems. Before she started
with two–digit she wanted to be certain that her students felt confident along with concrete knowledge and examples in their one digit equations
because if not, they would fall apart for two digit equations. My teacher had opened with two–digit addition, by developing knowledge on base ten
blocks for approximately two weeks, then progressed her students to the standard algorithm. My classroom is set up for their numeracy portion of the
class that... Show more content on Helpwriting.net ...
My student at the beginning of two–digit addition struggled with being organized in his base ten block drawings and would get the answer wrong
because he would circle 11 ones instead of 10. When he got the opportunity to work with the actual base ten blocks he had a strong concept of
regrouping. Before the lesson my cooperating teacher taught them how to do the standard algorithm, which my case study student took to excellent.
During the process of the standard algorithm I asked him to walk me through what he was doing and the process. He explained that he carried the one
over because it was making a new ten and then counted twenty plus thirty plus ten instead of 1 plus two plus three, which I felt like the was
understanding the place value. I thought during the lesson if my student was advancing ways to challenge him on this assignment would be to have
him add three two digit numbers or have him add one two digit and one three digit number. Then if I felt like my student was starting to struggle I
could have his partner help him solve the process, bring out the base ten blocks, or have him just add one two digit and one digit numbers until he got
the concept

Additional Mathematics Chapter 1 Essay
Chapter One From chapter 1, I feel a very important contribution to the development of mathematics is the Egyptian "doubling" method. The Egyptians
came up with this dyadic method when they realized that any integer can be written as a sum of "doubling numbers" or powers of two without
repeating any of them. By doubling a number enough to add up to the value of another, the Egyptians came up with an approach to multiplication that
we should purpose today. It is also an early introduction to the distribution property that will be used in later mathematics such as Algebra. I find this
method of "doubling numbers" as a path to multiplying large numbers that avoids the mistakes that can be made with multiplication in our current
method such as "dropping the zeros for the next place value"; and I will surely teach this method to my middle school students as they progress to
mastering mutiplication.
Chapter Two The "Babylonian" formula from chapter 2 is what I feel is a very important contribution to the development of mathematics. They
Babylonians came up with the formula and they did not even know it. ... Show more content on Helpwriting.net ...
Interesting that we actually get introduced to this theorem in our text earlier in time in the Babylonian Era of chapter two. Plimpton 322 actually
contained pairs of numbers in which the square root of the difference of the pairs' squares was a whole number; what we know as today as
Pythagorean Triples. But it was Pythagoras who took this property beyond just a concept and applied it to right triangles; where the sum of the two
squares of the two shorter sides of a right triangle equals the square of the hypotenuse. There is no way that I will teach mathematics and not come
across teaching any of my students the Pythagorean Theorem; whether it be Pythagorean Triples, right triangles, or Trigonometry, the Pythagorean
Theorem will be

The Impact Of Arabic Numerals In Medieval Europe
Impact of Arabic Numerals on Medieval Europe Medieval European society was changed by the introduction of the Arabic numerals into their society.
The Islamic Golden Age introduced lots of innovative thought into the world, and eventually those ideas made their way into Europe, one of which
was the Arabic numerals. They revolutionized the way that daily tasks, like merchant bookkeeping, and academia were approached. Medieval Europe
was transformed by the Islamic Golden Age and that is highlighted through the transformation Arabic numerals had on society. Preceding the Islamic
Golden Age, Indian culture had a revolution of thought which was seen in the Islamic Empire. One thing from Indian culture that transcended into
Islamic culture was the concept of zero. This was something that was not considered in earlier mathematic studies. It read in "Math Roots: Zero: A
Special Case," "the Arabs recognized the value of the Hindu system, adapted the numerals and computation, and spread the ideas in their travels." The
Arabic people saw the power in this numbering system because there was a place holder number. This concept was accepted into Islamic thought;
however, it was not received well in Europe. For the greater part of the European society, it was a strange system, in comparison to the Roman numeral
system, and was not widely accepted. At the beginning of Arabic numeral introduction into European society, scholars and mathematicians were
primarily the only ones who accepted

Essay On Ello
Step 14: Draw place–value blocks
Tell students, another way we can solve this is drawing place–value blocks. Tel students a line will represent a tens rod and a unit cube will be
represented by an X. (Draw and label on ELMO for students to see)
Model how to represent 39 with place–value blocks.
Draw the three long lines on ELMO to represent the three equal rows that Nelly wants to use to divide her 39 stickers.
Inside each line start to draw one vertical line to represent a tens rod, then draw 3 X's per line to represent 3 unit cubes.
Each "row" should have one line (tens rod) and 3 X's (three unit cubes).
Ask students how many place–value blocks do we have per row? (13)
"You can draw place–value blocks to help you solve division problems whenever you don't have place–value blocks to help you solve a problem. ...
Show more content on Helpwriting.net ...
Step 16: Write an equation
"Now that we have modeled division with place–value blocks and drawings lets come up with an equation."
Write on ELMO:
Step 17: Introduce mnemonic: "Does McDonalds Sell Cheeseburgers?"
Explain to students the mnemonic "Does McDonalds Sell Cheese Burgers?" will help them remember the steps for division
Write down these steps next to the division equation on the ELMO and go over each one with students o D– Divide (Г·) o M– Multiply (x) o S–
Subtract (–) o C– Check (пѓј) o B– Bring down (пѓў)
Provide handout to students for them to glue in their math

Conformity In Emerson's Self-Reliance
In the dictionary, conformity is defined as "compliance with standards, rules, or laws.". The reason I believe that conformity is a common element in
both Emerson's Self–Reliance and Gladwell's Outliers is because in Self–Reliance, Emerson believed that to be self–reliant, you must avoid conformity.
Gladwell agrees with this in chapter 8 of Outliers, where he states that we tend to conform to rules based off of where we are from and how we were
raised. This chapter was a perfect example because of how all those children were raised. As stated previously, a big key factor in Self–Reliance was
the concept that Emerson truly didn't want others to follow conformity. He wanted people to remain true to themselves due to the fact that it follows
along with the essence of the Universal Spirit. "To believe your own thought, to believe that what is true for you in your private heart is true for all
men,– that is genius" (Emerson, 19). After Emerson discusses conformity, he delves into the discussion of innocence and infants. He discusses this as
a nod to nonconformity. "Infancy conforms to nobody: all conform to it, so that the babe commonly makes four or five out of the adults who prattle
and play to it" (Emerson, 20). In a whole, Emerson's Self–Reliance was more or less

Description And Description Of A Calculator
<!DOCTYPE html> CALCULATOR body{ margin: 0px; padding:0px; } main section{margin: 20px;} #calc_body, #calc_header{ background
–color:
#BFBABE;} main section table {padding: 20px; border–radius: 10px; border–spacing:0px; border–collapse:collapse;} table, th, td
{display:inline–block; padding:0;float:left;} #heading{ text–align: center; font–size:23pt; color: black; font–weight:bold; text–transform: uppercase;}
main section table tr td button{ height: 70px; display:block; width: 90px; font–size: 22pt; border:0; } .longbutton{ height: 70px; width: 180px; margin:
0; text–transform: uppercase; font–size: 22pt;} main section table tr td button:hover{ background–color: #E990C0; font–weight: bold; } #display
{width: 540px; height: 80px; background–color: #B0EEDB; font–size: 25pt; border:none; outline:none; text–align:right; margin–bottom:15px;} #expr{
width: 540px; height: 80px; background–color: #B0EEDB; font–size: 13pt; border:none; outline:none; } #Trig{font–size: 26pt;}
.orangebuttons{background–color:#F99B58; } .orangebuttons:hover{ background–color:#FBB888;} .operators{ background–color:#7B8EF8;}
.operators:hover{ background–color:#B6C0FA;} #equalsign{background–color:#343436; color:white;} #equalsign:hover{
background–color:#959699;} Calculator log2 Rads

Questions On Learning And Teaching
Task #4: Mathematics Learning & Teaching
Competency 662.1.4: Aligning Learning Activities to National Standards
Competency 662.1.5: Standards and Best Practices in Teaching and Learning
Competency 662.1.7: Differentiated Instruction
Jennifer Moore
Western Governor's University
Part A: The "Equivalent Fractions and Decimals Lesson Plan" is aligned to NCTM's content and process standards. The content standard that this lesson
is addressing is numbers and operations. This entire lesson is about students using fractions and decimals to solve problems. This lesson also has
several process standards addressed in the lesson plan. One of the process standards used in this lesson is Connections. Throughout this ... Show more
content on Helpwriting.net ...
"The Show or the Ad" activity has students using word problems that talk about television programs and commercials. The teacher has made the
students connect that math is even in the TV shows that they watch. The teacher then makes it even more the students own connections by allowing
them to be creative and writing their own word problem about TV and commercials. The teacher also uses these types of problems on the summative
assessment. The teacher uses connections even in the homework that is assigned to the students. The teacher assigns students to watch their favorite
television show, and as they watch the show to record how many minutes the commercials are. All of this shows the students understanding of the
concept while allowing the students to relate this to something they enjoy. The one NCTM process standard that I would add to this lesson plan
would be the reasoning and proof standard. This is the only standard that is rarely used in this lesson plan. "The Reasoning and Proof standard
emphasizes the logical thinking that helps us decide if and why our answers make sense" (Walle, p. 5). This lesson needs to allow students to justify
the reason for their answers. This lesson only has them justifying their answers on one question, throughout the entire lesson. I would add more to the
questions than just the answers. One part of this lesson I would change to incorporate the reasoning and proof standard is on the "The Show or the

Children 's Knowledge And Understanding Of Place Value
EDP243 Children as Mathematical Learners Assessment One Child Study – Diagnosing Children's Knowledge and Understanding of Place Value
Tayhla Wood 18866371 Tutor: Kerrie Maxwell The following child study is divided into two components discussing aspects of place value as a concept
taught in mathematics education. Component A introduces the topic by discussing the significance of interview style assessment, followed by
discussion on the importance and teaching of place value. Misconceptions and difficulties surrounding place value learning are also described.
Component B focuses on an individuals understanding of place value through the use of an interview and tutoring sessions. 1. COMPONENT A 1.1
Rationale Essential aspects that underpin the professional and dedicated educator include the revising of knowledge and experience, reflection, and an
effort in understanding their students. Within mathematics, these skills are informed by the curriculum chosen, the students involved, and the
pedagogy that is selected, that create the professional judgement cycle (as seen in Appendix One) (Department of Education and Training Western
Australia [DETWA], 2013a). The more teachers understand about their students, the more they can adapt the learning environment to cater for these
different learning approaches (Burns, 2010). Assessing is a crucial component of education that informs teachers on individual development and
understanding (Booker, Bond, Sparrow

Common Sense Abbreviations
The common sense approach to abbreviations should apply to PSI reports as well. Of course one can abbreviate suffixes (Dr. Mr., etc.), words like
rd. (road) st. (street) and ave. (avenue). Agency names could be spelled out completely and followed in parenthesis, i.e. State Bureau of Investigation
(SBI). Beyond these, I see no reason to include abbreviations to the point that they are even a topic of discussion. Probation officers are not really
relaying a personal opinion in the PSI. They present documented and measurable progress of the individual under supervision. The officer might have
assigned a rating for the offender based on their behavior during an interview, such as level of remorse on a Likert scale. The recommendations

Earliest Civilization is the Region of Mesopotamia Because...
A civilization is recognized as such by its form of written language. For this reason, the earliest civilization is recognized in the region of
Mesopotamia with their language of Cuneiform. This ancient form of written language was inscribed on clay tablets that still remain in tact and are
being salvaged hundreds of thousands of years later. Even more impressive than just writing the language, however, is the ancient Babylonians' early
mathematical discoveries. These were also recorded with cuneiform and recorded on clay tablets, and like the language, served as an early
interpretation of mathematical principles that influence arithmetic all over the world today. Dating back to the second and third milennia BC,
Babylonians were so... Show more content on Helpwriting.net ...
The Babylonians used pre–calculated tables to assist with arithmetic. Perhaps the most amazing aspect of the Babylonian's calculating skills was their
construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the
numbers up to 59 and cubes of the numbers up to 32. Most frequently Babylonians utilized tables of squares and cubes to simplify multiplication. The
concept of reciprocals was also first introduced by the Babylonians. Because they did not have a method for long division, they were able to recognize
that using their sexiagesimal system of numbers, numbers with two, three, and five, had finite factors of which tables have been found. For numbers
not containing one of the finite factors, the Babylonians used approximation reciprocals. The pre–calculated tables method is also how the Babylonians
incorporated algebra in their number system. They were the first people to use the quadratic equation, though not in its exact form. They used the form
x2+bx=c which, when solved, can be interpreted as x=–b/2–в€
љ(b/2)+c which more closely resembles the modern quadratic equation. Using their
arithmetic tables of squares, the Babylonians were able to interpret them in reverse to find square roots. Because everything was a real problem, they
always used the positive root when solving. Most commonly squares were used for finding

Why Is A Manipulative?
A manipulative is often used in many ways to teach mathematics such as basic addition, fractions, decimals, order of operations. To name a few
manipulatives; blocks, cards, number tiles, counting tubes, etc...A manipulative can be taught either concrete (hands–on) or virtual. Hands–on
manipulative models are physical objects such as base–ten blocks, deck of cards, Dice games, and Algebra tiles. A virtual manipulative is a technology
that models the existing manipulatives such as base ten blocks, rulers, fractions bars and algebra tiles to name a few. These manipulatives are in the
form of Java or Flash applets, a web base technology. Normal playing cards have so many uses in teaching mathematical lessons. When teaching your
students how to add, subtract, sort, compute or compare numbers, using a deck of cards gets the students working with numbers while playing a
game. There are two advantages of using a deck of cards as a hands–on manipulative: First advantage; cards are inexpensive, can be easily put away.
Second advantage; the cards can be used as independent practice or cooperative learning. During cooperative learning the students become involved &
they are able to practice the edification adeptness such as multiplication games, fractions, percentages and decimals. There are two disadvantages of
using a deck of cards as a hands–on manipulative; often students end up dropping things on the floor which becomes a disadvantage because you
spend time making sure

The History of Zero: Indian and Mayan Cultures
Zero is usually recognized today as being originated in two geographically separated cultures: the Maya and Indian. If zero was a place–holder symbol,
then such a zero was present in the Babylonian positional number system before the first recorded occurrence of the Indian zero. If zero was
represented by an empty space within a well–defined positional number system, such a zero was present in Chinese mathematics a few centuries before
the beginning of the Common Era. The absence of a symbol for zero in China did not prevent it from being an efficient computational tool that could
handle solution of higher degree order equations involving fractions.
However, the Indian zero was a symbol, a number, a magnitude, a direction separator and a place–holder, all in one operating within a fully established
positional numeration system. Such a zero occurred only twice in history – the Indian zero which is now the universal zero and the Mayan zero which
occurred in solitary isolation in Central America at the beginning of the Common Era. To understand the first appearances of the Indian and Mayan
zeroes, it is necessary to examine them both within the social contexts in which both of these inventions occurred. Because of the popular difficulties
with the zero, there has occurred over time a series of avoidance mechanisms to cope with the presence of zero. The word zero comes from the Arabic
meaning void or empty which became later the term for zero. The ancient Egyptians never

The Binary Number System In The Hindu And Boolean Algebra
In the modern world, binary numbers have great importance in computer coding as well as in Boolean algebra Without the application of binary
numbers, computers would not be able to exist or communicate with each other. Binary numbers include a base two number system rather than the
base ten system which is used in a math class such as algebra. A base two system means that the only numbers that are used in that system and in
binary those numbers are 1 and 0. A base ten number system beans that numbers 1, 2, 3, 4, 5, 6, 7, 8,9, and 0 are used. All numbers can be converted
from decimal, the base ten set of numbers, to binary and from binary to decimal allowing computers to communicate. "The Hindu–Arabic system is
based on ten different symbols and is considered to be a base 10 system. Numeral systems with different bases have found use in applications where a
different base provides certain advantages" (Lande 514). The Hindu–Arabic system is the decimal system that is used during a math class and the
binary number system is used for different functions in computers or Boolean algebra. Due to the Hindu–Arabic system and the binary number system
have different numbering bases, they must be converted to the other. The binary number system is a base two system which has to be converted to a
base ten if it was to be expressed in decimal, the Hindu–Arabic system. The number 10011011 in binary would be 155 in decimal. This process would
be done by writing the binary number then

History Of Roman Number Symbols
The history of Roman Number Symbol are represented by letters. The Roman numerbs are represented by seven different letters are I, V, X, L, and D.
Therefore, these roman letters represent 1, 5, 10, 100, and 500. Ancient Roman use these seven letters to make a lot of different numbers and to be
written of the Roman alphabet. In the Etruscans was an ancient civilization of Italy developed their own numeral system with different symbols. A
common theory of the origin Roman numeral system was represented by hand signal. For example, the Roman numeral system by hand signal was
used like one, two, three and four signaled by the equivalent amount of fingers that were used. Then, "The number five is represented by the thumb and
fingers separated, making a 'V' shape and The number ten is represented by either crossing the thumbs or hands, signaling an 'X' shape" (Pollard).
Therefore, the numbers; six, seven, eight and nine are represented by one hand signaling a five and the other representing the number 1 through to 4.
The hand signal was used for counting by either crossing the thumbs, fingers separated, and signaled, which helped to hand ... Show more content on
Helpwriting.net ...
The tally sticks had been used for thousands of years and continued to be used until the 19th century. For instance, the tally sticks was used to either
additive nor subtractive, whereas the numbers one, two, three and four were represented by the equivalent amount of vertical lines. If these
numbers described in the article would be written in tally sticks a Roman numerals. For instance, "Four could be written as either IIII or IV"
(Reddy). Another example, seven on a tally stick would look like, IIIIVII, when shortened it would look like VII. These Roman numbers are the
same like the Roman number symbol. Another reason, larger numbers in tally sticks like 500 and 1000 would be a 'D' and 'M' in a circle

The Learning Of Mathematics And Mathematics
Overview In 21st century classrooms, an educators teaching practice is vital in developing student's mathematical knowledge. A constructivist
approach is required to allow students to use their prior knowledge to make sense of new information through hands–on activities. To effectively equip
students with the necessary skills to see connections in mathematic concepts, Big Ideas must be employed by educators. The Australian Curriculum
supports the use of Big Ideas to deepen students understanding of mathematical content. Students learn effectively when they can see connections
between concepts. The most effective way to link an array of mathematical concepts is through Big Ideas. A Big Idea is defined as a "statement of an
idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole" (Charles 2005). Big
Ideas are broken down into several elements. These are the name of the Big Idea, the idea central to the learning of mathematics and the links to several
mathematical understandings. Students need to learn the mathematical understandings to allow them to understand the Big Idea (Charles, 2005). Big
Ideas are important as they should be the basis for student learning, the teaching pedagogy for educators and encompass the Mathematics curriculum.
Hiebert et al. (as cited in Charles, 2005) agrees and explains that individuals understand concepts if they can see a link to other things they know, which
is the

The Numbers And Counting System In Ancient Egypt
Numbers and counting systems are used daily by everyone. Numerous civilizations throughout history like Egypt, Babylonian, Maya and Africa
developed a unique number system. They were commonly used to communicate numbers in an everyday life. Since then counting has changed. A few
of the earlier systems had principles that survived and helped in some way, shape or form to create our current Hindu Arabic numbering system. It has
become widespread. Back in 3000 B.C, ancient Egypt was using the number system to calculate areas of land, distribute money, and much more. The
Egyptian counting system consisted of hieroglyphs and pictorial signs. The graphic numbers can be of a person, animal, or plant. The hieroglyphic
numbers were a written form of the number system. It used a decimal base approach. For all the powers of ten, there was a unique symbol. For
example, the number ten was represented as a upside down "U", for the number a hundred thousand it was tadpole, for a million it was represented by
a man kneeling with his arms raised. When reading or writing the number, it is written from right to left. It was written on temples, vases, and stone
monuments. The Egyptians used hieroglyphic numbers to advance in architectural achievements. Math was used to make historical creations like
tombs and pyramids. Greenwald and Thomley agree that "the ancient Egyptians were also aware of fractions, which were primarily written as unit
fractions of the form 1вЃ„n, such as 1вЃ„2or 1вЃ„4..."

Examples Of Josephus Problem
The next base the essay will explore is base 3 also known as ternary, this is the base that was used in the original Josephus problem. By exploring this
base and a few more bases, I will find a pattern and derive an equation to solve Josephus problem with any number of people in any number base.
Josephus problem in base 3 is when each person kills the person who sits two seats away from him and the following diagram will demonstrate how it
happens, the numbers on the inside show the order of the people that are gone which will make it easier to follow.
Figure 7 An example of how Josephus problem works in base 3 with 12 people
I will now create the same table as I did for the problem in a binary form.
Table 3 Analyzing the Josephus problem in base 3 when1в‰¤nв‰¤12
Number of people
()
Order in which people are gone
The winning number
1
1
1
2
1,2
2
3
3,1,2
2
4
3, ... Show more content on Helpwriting.net ...
Applying the equation to Josephus problem, to find we have to find as . We know how to solve as there are only 2 people and starts meaning he has to
skip and therefore dies first and wins. Now we know that , we can see from the table above the answer is correct however we have to do it once more
to find .

We can always consider position and we have to repeat the equation until getting to the required number. This solution can be very frustrating as you
must do the same thing over and over, however it can be programmed very easily and save a lot of time.
This method is the third and the last method to solve the Josephus problem, it is the most useful method as it works for all bases however it is the
most complicated one. In order to compare between this method and the other two methods, we first have to write the equation in base

Justifications : Warm Up : Shared Book Essay
Justifications: Warm–up: Shared Book Description:Justification: The warm up for this lesson is reading a book called, "Earth Day – hooray!" I chose
to incorporate reading as a math warm–up because it is an effective way to explore mathematical concepts, which are both engaging and informative for
the students. Integrating elements of literature into the lesson supports problem solving and allows students to be fully immersed in the story, solving
themselves the mathematical issues that the characters fall into. Ministry of Education, 2016, endorses this idea stating that applying visual elements
of picture books aid in illustrating mathematical concepts, which are difficult to teach with other resources, or challenging for students to comprehend.
This particular book introduces properties of place value, discussing the idea that 10 tens is equal to 1 one hundred, which all groups need assisting
with. Picture books contextualize mathematical ideas in meaningful ways for students, while creating stories, which are relatable and spark curiosity
and enthusiasm (McGrath, 2014). McGrath further explains that learners will respond positively when stories are not written with the intention of
teaching children mathematics and will be more receptive to learning the math message, if the thinking is second nature. As the cappuccinos are
learning about place value this warm–up is a great launching point for furthering exploring this concept and prompts ideas for discussion and

Mathematics and Theology Blossoming Together
Mathematics and theology have blossomed together throughout history with many great mathematicians also being great theologians. However, in the
modern scientific era, mathematics has become by and large secularized in mainstream academia. Although the secularization of mathematics seems to
ignore mathematics' metaphysical value, in truth, this secularization allows for mathematics to act as a universal tool and allows the individual to
attach his or her own personal truths without marginalizing the beliefs of others, especially in education. In addition, attaching one's personal beliefs to
systems of math and logic may lead to contradicting interpretations of the material when taken into the larger context of society, such as with the
concept of infinity, the meaning of truth and proof, or even the source of mathematics itself. In essence, the secularization of mathematics is a necessity
in our modern dynamic world and in order for mathematics to maintain its effectiveness as a universal tool our personal beliefs must not be attached to
mathematics as a whole.
To begin with, a proper definition of secularization is necessary in order to establish the correct connotation for the term. As denoted by the world
renowned scholar in the study of the sociology of religion, Jose Casanova, secularization is spoken of in three senses:
a.)Secularization as the decline of religious beliefs and practices in modern societies, often postulated as a universal, human, developmental

Why Do Lincoln Use Significant Figures
Any good chemist would understand that when Abraham Lincoln said "four score and seven years ago" he was actually using significant figures. The
Declaration of Independence was signed on July 4th while Lincoln delivered this speech on November 19th. This means that it actually occurred four
score, seven years, four months, and 12 days before. Admittedly that does not roll off the tongue as smoothly, so it's understandable that Lincoln would
only use two significant figures. Significant figures are the reason that the beginning lines of the Gettysburg address will forever be remembered by
Eighth graders across the country. Without the use of significant figures, numbers in chemistry wouldn't have any meaning. I could say 2g when I'm
really ... Show more content on Helpwriting.net ...
If you are given a measurement of 4g and a measurement of 4.0001g, you would understand that the equipment used to measure 4.0001g is much
more precise than the equipment used to measure 4g. Although significant figures tell us about the precision of the measurement, we really cannot
attest to the accuracy of the measurement unless we are sure of the correct answer. The same rule about the estimated digit is applied when taking
measurement. The way to report an analog measurement is to write down all the digits you are "confident" in and estimate one digit further. For
example, if using a 10ml graduated cylinder with nine markings in between the numbers 1ml and 2ml and the liquid falls between 1.5ml and 1.6ml,
but closer to 1.5ml, the digits you are confident in are 1 and 5. Therefore you estimate another digit which you think it could be such as 1.53ml. When
making a digital measurement, the rule is to always write down every digit while understanding that the last digit is an estimation. There are also a few
important differences between the course policy and the textbook. One is that the book states that if you place a $#$#0$#$# to the right of trailing
zeroes, it would make those zeroes significant e.g. 500. has 3 significant figures while 500 has 1. This is not true and they should be ashamed. Another
difference is in the convention of marking significant figures. In this course, Professor Halpin will underline all significant figures to identify them. It
just makes it easier to get the correct answer while rounding to the correct number of significant figures. The book does not do

Why Is A Manipulative?
A manipulative is often used in many ways to teach mathematics such as basic addition, fractions, decimals, order of operations. To name a few
manipulatives; blocks, cards, number tiles, counting tubes, etc...A manipulative can be taught either concrete (hands–on) or virtual. Hands–on
manipulative models are physical objects such as base–ten blocks, deck of cards, Dice games, and Algebra tiles. A virtual manipulative is a technology
that models the existing manipulatives such as base ten blocks, rulers, fractions bars and algebra tiles to name a few. These manipulatives are in the
form of Java or Flash applets, a web base technology. Normal playing cards have so many uses in teaching mathematical lessons. When teaching your
students... Show more content on Helpwriting.net ...
A disadvantage of using dice in a classroom can be a bit noisy with the rolling of the dice & students possibly dropping them on the floor. Although
Dice games can be used to teach multiple math concepts such as learning multiplication, addition and subtraction and decimals to name a few the
games can tend to get boring. The students may either get board or just play with the rolling of the dice for fun. Base Ten Blocks has been the most
used manipulative numerical system in just about every classroom all over the world. They help elementary students visualize the base ten system
while working on basic or a more advance math lesson. One advantage of using Base Ten Blocks; when introduced early in elementary education it
helps students learn to develop their counting skills at a quicker pace and provides a solid strong foundation for advanced mathematics. Another
advantage of using Base Ten blocks; they are great to use in guided math groups. Students are able to develop problem solving skills while working
together in groups. A disadvantage of using Base Ten blocks; if there is just teacher demonstration and no hands on learning the students will not grasp
the concept so readily. Another disadvantage; to multiply is that any product over a certain size cannot be represented (2002, M Freitag) Algebra math
tiles help students of all learning styles to better understand mathematical ideas. One advantage of using Algebra tiles; they are

Comparison Between Roman And Modern Era
From calculating digital computations to setting the foundation of modern applications, the number system is more relevant in life and culture than
most people realize. For example, people use the number system to organize sections and chapters from literature, keep financial affairs orderly and
neatly, and compute numerous types of everyday costs. Also, it dictates how modern applications function and how programming languages work
seamlessly. Despite these incredible accomplishments, the world would lack all knowledge of it were it not for ancient number systems. These systems
include three of the most famous ones in the world: Roman numerals, Mayan numerals, and Egyptian numerals. Through innovation, technology, and
ingenuity, ancient ... Show more content on Helpwriting.net ...
For example: IV = 4, IX = 9, XXXIV = 34, CIX = 109, CD = 400, and MCMIV = 1904. The key to handling this exception is to subtract the digit ahead
of the power from the numeral itself [4].
Now that the basic numbers and rules have been covered, the arithmetic for addition, subtraction, multiplication, and division can be understood. To
calculate addition, one must first rewrite the numbers to arrange them in descending order after putting them together. Next, one must learn to ignore
the subtractive system and write out individual numbers when appropriate. For example: 4 = IIII, 9 = VIIII, 34 = XXXIIII; but 3 still equals III, 6
still equals VI, and 37 still equals XXXVII. Third, evaluating in ascending order, one must rewrite trailing "I" digits into a simplified version. Finally,
any further required simplification is performed. To demonstrate this, consider two numbers in addition, 13 + 66. The numbers are written side–by–side
(XIII + LXVI), then put together in descending order (LXXVIIII). The four trailing "I" digits are rewritten as IV, and this number is rewritten as
LXXVIV. The last three digits, the "VIV," is equivalent to "IX (9)," so the number is rewritten one final time as LXXIX (79).
For subtraction, one must borrow numbers by marking out common symbols in the second number. For example, LXVIII – XII (68–12) =

The Importance Of Student Achievement
Student achievement is a multi–faceted concept and does not look the same for every student. At the core of student achievement are comprehension
and connection. Students must be able to understand and contextualize instructional material to foster comprehension. Equal to comprehension is the
ability of students to connect with the material and make relevant associations between concepts and skills for application. The intention of instruction
based on prioritizing and unwrapping standards helps educators to define the instructional pathway that best supports the standard and facilitates
student learning. Learning pathways align essential objectives, activities and resources to outline progress within a standard. Learning pathways detail
the skills students need to acquire for demonstrating comprehension and mastery, which constitute achievement. Additionally, learning pathways
present an interpretive value for gauging student understanding and performance. Gutierrez explains that learning pathways "must have strategic
opportunities to build in prior knowledge, reflection, and application" (Gutierrez, 2017, para 8). Gutierrez further positions that "learning pathways are
intended to be flexible, multidisciplinary, and increasingly personalized" (Gutierrez, 2017, para 8). Without properly considering and using learning
pathways, students are not provided opportunities to connect content with comprehension and skill. Effective learning pathways scaffold on what
students

Ancient China Research Paper
In 1,500 bc, the mathematicians of China were using numbers. Long before the people of India invented the number zero, Chinese people were using
base ten counting systems. The base ten method they used is different than the one we use today, using ancient Chinese math, the number 465 would
be written as four one hundred markers, six ten markers, and five one markers. They were also advanced in geometry, this was proved in 600 BC,
when the Tangram was invented. The Tangram is an amazing math game, that allows you to explore geometry to huge extents. This essay tells the
fascinating story of how the game is played, how it was used, and how it became known to the western world. China is often thought of as one of the
wealthiest countries in... Show more content on Helpwriting.net ...
In ancient times, believe it or not, they were used to tell stories. The pictures made by Tangrams can be put in order to make a scene. You can
arrange the shapes in any way, and trace the picture onto paper, and make a new figure. The traced pictures can look just like comics. You actually
can think of Tangrams as the comic books of the ancient world. Quite truly, you can make any story, from superheroes to mathematics. This is
actually quite important, creating a story using geometric shapes can actually improve your skills in geometry. That probably explains why they were
so popular in China. "Scholars and artists thrown together are often annoyed at the puzzle of where they differ. Both work from knowledge; but I
suspect they differ most importantly in the way their knowledge is come by. Scholars get theirs with conscientious thoroughness along projected lines
of logic; poets theirs cavalierly and as it happens in and out of books. They stick to nothing deliberately, but let what will stick to them like burrs
where they walk in the fields." –Robert Frost. This quote highlights the fact that both scholars and artists take an interest in Tangrams. "Tangrams are a
great thing to incorporate into the mathematics classroom because they are fun, interesting, and

Thinking, Fractions And Decimals
Teaching students effectively in areas of multiplicative thinking, fractions and decimals requires teachers to have a true understanding of the concepts
and best ways to develop students understanding. It is also vital that teachers understand the importance of conceptual understanding and the success
this often provides for many students opposed to just being taught the procedures (Reys et al., ch. 12.1). It will be further looked at the important
factors to remember when developing a solid conceptual understanding and connection to multiplicative thinking, fractions and decimals.
When teaching mathematical concepts it is important to look at the big ideas that will follow in order to prevent misconceptions and slower
transformation ... Show more content on Helpwriting.net ...
C., 2015, p. 3).
Symbolic representation using base–ten and expanded algorithms is a way to show students the written connection to the visual models used. The
partial–products algorithm is a more detailed step–by–step process and therefore more advisable to avoid errors in students learning to grasp the
procedure (Reys ch.11.4). This process allows students to visualise the distributive property more easily. However, the standardmultiplication
algorithm is quicker and acceptable for students, if the teacher feels they have complete understanding of the steps in the partial–products algorithm.
Multiplication by ten gives students opportunity to explore larger numbers, and can also be extended on(Reys et al. ch. 11.4). In addition, multiples of
10 give students the knowledge that all digits move left one place and an additional place hundreths. This concept can be used to introduce the decimal
place which is also moving place each time something is multiplied by tens. Exposing students to a range of examples which displays patterns that
occur when multiplying by tens and hundreths will generate meaning of digits moving place (Reys et al., ch. 11.4).
Visual models known as arrays or grids can be introduced early to assist students thinking by providing a visual representation when going from adding
to multiplying. In addition, arrays are a great

The Linguists: Boarding Schools
The Linguists Linguists study language because they are concerned about the amount of languages disappearing. In the video it states, "One language
is lost per two weeks." When a language dies, some kind of unique world could be lost. Gregory and David are visiting places that are most in need.
These places tend to be smaller in population. Younger children are typically the ones to stop speaking the indigenouslanguage. I believe since they're
language is not as popular and most these languages don't have a written form, it would be easier to learn a different language for learning purposes.
Maybe schools are the ones who don't teach these indigenous languages to them. Boarding schools in some of these places bring in groups of
minorities of tribes to a central location to teach them. In the video, the boarding school in India brought in 60 tribes around the location to teach
them English, and the practice of Hindu religion which presents subtle pressure on the children's identity. The learn many valuables skills at
boarding schools but bringing in of many different types of groups affects their learning about their origin tribe or group. When the kids go home
they do not speak their origin language as much or even at all. Older generations tend to be the only fluent speakers of these dying languages.
Parents even in their fifties or sixties are considered to not speak their origin language because this way of education started around their time.
There is a way to resolve this conflict, and that is if every indigenous group had a school of their own. This would be a solution but many of these
places can probably not afford a school for their indigenous group. There is a solution but it is a question if the solution could be realistic for the
communities or not. Since most of the tribes are not Hindu yet they're learning Hindu at the school. This may show up in census of religion in areas not
being of their origin religion but as Hindu. Younger children's may mostly convert to Hindu because that is what they are taught in school thus making
their indigenous group's language one step closer to being extinct. Elicitation is a technique of getting any of a number of data collection. This relates to
the film

Essay On The Babylonians
The Babylonian civilization existed from around 3000 BC until 539 BC. The civilization has its roots to Mesopotamia, a plain between the Tigris
River and the Euphrates River, which is now in modern day Iraq. The Babylonians are notoriously famous in their discoveries and inventions, most of
which we still use today. The civilization–developed ideas such as astronomy, in which they created the Saros Cycle, a cycle used to predict solar and
lunar eclipses. The Babylonians were also masters in architecture, constructing buildings like the Hanging Gardens using innovative techniques
including glass–covered bricks and adornments of gold and bronze. However, despite many of these discoveries being forgotten or lost, their
advancements in mathematics have been kept for centuries, and have brought modern ideas of angles, quadratic equations and even knowledge about
the Pythagoras theorem, before Pythagoras himself existed.
One modern–day use of mathematics and quadratic equations, created by Babylonians, was the collection of taxes. Mesopotamia was a very fertile land;
therefore there were a high number of farmers in the Babylonian Empire. Because of the high number of farmers, tax collectors and mathematicians
had to develop a new and more efficient way of calculating the ... Show more content on Helpwriting.net ...
Despite the fact that many of their discoveries have been lost through time, it can be concluded that the Babylonian Empire was very advanced for its
time and despite ending with a societal collapse, it is still possible to find everyday things without knowing they possess century–old roots connecting
to the ancient Babylonian civilization. This truly shows, the large influence the Babylonian civilization has on our modern day society and our
understanding of the

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