Tuesday, 30 June 2009
Ten science projects for kids
A listing of science project for kids to do for fun or for a Science Fair.
Whether you're in search of a science project for a school fair or simply wanting to experiment at home for fun, you'll find something below to fulfill your needs.
Air Stuff
You'll need:
Cardboard or stiff poster board
Six 1 1/2 inch x 3 1/2 inch sticky surfaces (Transparent tape works great!)
Six 12 inch long pieces of string
Tape
Hole puncher
A magnifying glass
Cut out six air particle collectors from the cardboard or poster board. Be sure to make them large to cut an opening for the sticky surfaces in the center and still leave room around the edges for vital information. In the margins around the edges, write the words location, starting date, starting time, ending date, and ending time. Punch a hole in the top of each collector, then tie a string through the each hole. Affix the sticky surfaces to the center cut outs. Next, hang each collector in a strategic place. A few good ideas for where to hang them are by your bed, over the stove in the kitchen, outside under a tree, by an air vent, and inside or outside of a window. Wait for two to three days before taking the collectors down. Once down, inspect what air particles have accumulated on the stick surfaces. Dust? Pollen? Dandruff? Sand? Write down your findings and list which location had the most or the least amount of air particles.
How do animals spend the winter?
You'll need:
Popcorn, cranberries, orange slices, pear slices, apple slices, bird seed, peanuts in the shell, and peanut butter
Pine cones
String
Needle
Strong Thread
Scissors
Knife
Empty milk gallon, cleaned thoroughly
Cover pine cones with peanut butter, then attach with a piece of string. Thread cranberries and popcorn on one length of string and peanuts in the shells on another. When slicing the fruit, be certain to slice along the width so that the pieces will be circular in fashion. String the fruit on several strands of thread. Cut a large hole in the milk container and fill with the base below the opening with birdseed. Hang the winter goodies from trees in the yard. Be patient. It takes awhile for birds and animals to locate the food. Once they've found your treats, observe them and make notes about what you discover.
Why do leaves change color?
The object of this project is to separate the colors in a leaf using chromatography. Important: Be sure to have an adult help you!
You'll need:
Leaves
Baby food jars with lids (If the lids are absent, substitute with plastic wrap or aluminum foil.)
Rubbing alcohol
Coffee filters
Shallow pan
Hot tap water
Tape
Pen
Plastic knife or spoon
Clock or Timer
Collect 2-3 large leaves per tree from a variety of trees. Chop the leaves into small pieces ensuring not to mix the varieties. Place in baby food jars and label with the name or location of the tree. Add enough rubbing alcohol to cover the chopped leaves, then use a plastic knife or spoon to grind the leaves in the alcohol. Cover the jars with lids or substitute coverings loosely. Fill a shallow pan with one-inch hot tap water and place the jars in the water. Set a timer for 30 minutes. Give the jars a twirl every five and wait for the alcohol to become colored. It may take a little bit longer than 30 minutes for a full effect. The darker the color, the better. If the hot water cools, replace it. While you're waiting on the alcohol to become colored, cut thin strips from the coffee filter paper and label them with the names or locations of the trees. Remove the jars from the water and take off the lids. Slide a piece of filter paper into the jar so that one end is in the leave and alcohol mixture and the other is bent over the side of the jar. Secure the bent end with a piece of tape. The alcohol will travel up the filter paper. The process takes about 30 to 90 minutes. Possibly more time is necessary depending on the type of leaves. Once finished, you should be able to see the different shades of green in the leaves and there may be orange, yellow, or red present as well.
Edible Science
Build a salad using every part of a plant.
You'll need:
Spinach, lettuce, or cabbage (Leaves)
Carrots or onions (Roots)
Asparagus or rhubarb (Stems)
Broccoli, cauliflower, or pansies (Flowers)
Tomatoes, cucumbers, squash, melons, apples, or oranges (Fruit)
Peanuts, sunflower seeds, peas, beans, pumpkin seeds, or pine nuts (Seeds)
Knife
Bowl
Poster board
Markers
Draw a plant on the poster board and label each part with the appropriate plant name. Chop you preferred items from the list. Add to the bowl. Toss and present.
Interesting Plant Fact: There is only one plant that can be eaten whole. It's the bean sprout.
Allelopathic Observations
This project takes some time, but if you know you'll be needing a project for the school year or just want to experiment while you're helping your parents in the garden, it's a lot of fun.
You'll need:
Sweet potatoes with roots, ready to be planted
Other plants or seeds
Pencil and notebook
Plant sweet potatoes in increasing distances from other plants. Be sure to leave a few control plants with no sweet potatoes near them. Find out how fast or slow the other plants grow and make notes each week about your discoveries.
Why is the sky blue?
Refraction
You'll need:
A small mirror
Piece of white paper or cardboard
Water
Large, shallow pan
Direct sunlight
Fill the pan 2/3 with water. Place it in direct sunlight in order for the experiment to work. Hold the mirror under the water while also holding the piece of paper in the other hand. Adjust both so that light reflects in order to observe the prism color spectrum.
Bottled Sky
You'll need:
A clear glass jar
Water
Milk
Measuring spoons
Flashlight
A darkened room
Fill the jar with water 2/3 full. Add a 1/2 teaspoon to a teaspoon of milk, then stir. Use the flashlight to shine from the bottom, the top, and the side. Note the difference in colors each time you alter the light source. The effect is similar to what dust particles would do in the air causing varying shades in a sunset or making the sky blue. The effect is due to scattering of particles and changing light source.
Mixing Colors
You'll need:
Pencil
White paper
Ruler
Markers or crayons
Small bowl
Paper cup
Use the bowl to trace a circle onto a piece of white paper, then cut it out. With the ruler, divide the circle into six equal parts. Color as evenly as possible each section with one of the following colors: red, blue, orange, yellow, green, and purple. Poke a hole through the center of the circle with the pencil, leaving the pencil partially threaded through. Poke another hole in the center of the bottom of the paper cup. It should be a tad larger in diameter than the pencil. Turn the paper cup upside down on the piece of paper and insert the pencil through the hole. Adjust the colored circle so that it is approximately a 1/2 inch above the cup. Give the pencil a twirl and watch the colors on the circle. The colors on the circle are the main colors in white light. As you spin the circle, it should appear as though the circle is white.
Primitive Microscope
You'll need:
White paper
Ruler
Scissors
Pencil
Items to observe
Measure and mark a one-inch square on a piece of paper. Cut out the square. View items such as leaves, soil, flowers, and bark beneath the makeshift microscope. Take as much time as you can to observe what you can see. On another piece of paper answer a series of questions on your findings. Is it hot or cold? Is it soft or hard? Is it wet or dry? Is it bright or dark? Is it smooth or rough? Is it alive or not? How many colors do you see? How many shapes do you see? Is there anything moving? If so, what is moving? Why is it moving? Be creative and come up with your own questions.
Stages in the Life of a Butterfly
You'll need:
Scissors
Glue
Pipe cleaners
Toilet paper tube
Egg carton
Markers or crayons
Ice cream pop stick or tongue depressor
Heavy paper
Poster board
Pompon balls
There are four stages on the life of a butterfly: an egg, larva or caterpillar, chrysalis or pupa, and butterfly. Explain each stage on poster board with a visual aid for every stage. For the egg, cut out a section from the egg carton and color it. If you'd like to include a transitional stage, cut out a second section from the egg carton and glue two pompon balls coming out from beneath to make it seem as though the caterpillar is emerging from the egg. For the caterpillar, form a squiggling line with pompon balls. For the chrysalis stage, color the toilet paper tube black or green. Draw, cut, and color a butterfly onto heavy paper. You can make the butterfly look however you want, but be sure that its wings match. Punch a hole through the top of the butterfly. Make a "v" with a pipe cleaner, curling the ends. Slip the pointed tip through the hole and wrap around the base of the antenna. Glue the butterfly onto the ice cream pop stick. Once the glue is dry, you should be able to slide the butterfly in and out of the chrysalis easily.
Monday, 22 June 2009
Homeschool vs Public School
By J. E. Burke
Parents are starting to find out that one of the best ways to teach their children is though homeschooling. Amazingly, even children are choosing homeschool over public school. The benefits of a homeschool vs public school education are many. Children that are homeschooled have more time to work on projects, are able to have more hands on experiences and they don't have the negative peer pressure that exists in public schools. Many parents choose homeschool over public schools because they want to teach their children religion and moral values. In the debate over homeschool vs public school, homeschooling is winning in many states.
One of the primary benefits of teaching a child at home is designing a curriculum especially for them. Parents can spend more time on subjects that children are struggling with. It is estimated that an average 45 minute class time is spent in the following ways by a teacher: 15 minutes is spent on discipline problems, 15 minutes on teaching the subject and 15 minutes spent on paperwork. The time that a teacher can spend on a student during this time in a classroom of 25 is less than one minute per child. This means that the only children getting attention in the classroom are children with behavior problems. This leaves the majority of children struggling to understand subject material and not having the time to ask questions.
Many parents learn early on to accept the fact that their child is just average. The exact opposite may be closer to the truth. Your child may have skills and abilities that are never recognized in the classroom because the teacher doesn't have the time to get to know him or her. The typical public school is focused on an outcome-based education system. This means that children are taught to take tests. Many teachers are beginning to complain that it is no longer fun to teach in the classroom. Teaching to the students and not the task is what most teachers want to do. Instead, the individual states have taken away the creativity of teachers and they are no longer able to design individualized curriculums for their students. It is easy for parents to decide in this instance that the winner in homeschool vs public school options is homeschooling.
One final reason that parents are weighing the options of homeschool vs public school is that bullying fellow classmates is becoming prevalent. Many schools are having problems with teens designing websites on campus where they threaten and taunt their fellow classmates. The common use of cell phones has also caused problems such as harassment of classmates. Each year hundreds of teens commit suicide, many times because their peers are making their lives miserable. If your child is having these kinds of problems consider taking your child out of school for their own sanity as well as yours. Homeschool is a great education option and it provides a safe learning environment for your child.
To stay abreast of the current trends, information and resources available for homeschool parents, teachers and students subscribe to Homeschool Success News -
http://hsnews.homeschooltutorpro.com
If you need an online tutor, courseware or wish to offer your services as a paid online tutor contact our TutorBug website: http://homeschooltutorpro.com
Magic Learning Systems also provides excellent products to enhance the homeschool experience for teachers and students. http://magic.homeschooltutorpro.com
Parents are starting to find out that one of the best ways to teach their children is though homeschooling. Amazingly, even children are choosing homeschool over public school. The benefits of a homeschool vs public school education are many. Children that are homeschooled have more time to work on projects, are able to have more hands on experiences and they don't have the negative peer pressure that exists in public schools. Many parents choose homeschool over public schools because they want to teach their children religion and moral values. In the debate over homeschool vs public school, homeschooling is winning in many states.
One of the primary benefits of teaching a child at home is designing a curriculum especially for them. Parents can spend more time on subjects that children are struggling with. It is estimated that an average 45 minute class time is spent in the following ways by a teacher: 15 minutes is spent on discipline problems, 15 minutes on teaching the subject and 15 minutes spent on paperwork. The time that a teacher can spend on a student during this time in a classroom of 25 is less than one minute per child. This means that the only children getting attention in the classroom are children with behavior problems. This leaves the majority of children struggling to understand subject material and not having the time to ask questions.
Many parents learn early on to accept the fact that their child is just average. The exact opposite may be closer to the truth. Your child may have skills and abilities that are never recognized in the classroom because the teacher doesn't have the time to get to know him or her. The typical public school is focused on an outcome-based education system. This means that children are taught to take tests. Many teachers are beginning to complain that it is no longer fun to teach in the classroom. Teaching to the students and not the task is what most teachers want to do. Instead, the individual states have taken away the creativity of teachers and they are no longer able to design individualized curriculums for their students. It is easy for parents to decide in this instance that the winner in homeschool vs public school options is homeschooling.
One final reason that parents are weighing the options of homeschool vs public school is that bullying fellow classmates is becoming prevalent. Many schools are having problems with teens designing websites on campus where they threaten and taunt their fellow classmates. The common use of cell phones has also caused problems such as harassment of classmates. Each year hundreds of teens commit suicide, many times because their peers are making their lives miserable. If your child is having these kinds of problems consider taking your child out of school for their own sanity as well as yours. Homeschool is a great education option and it provides a safe learning environment for your child.
To stay abreast of the current trends, information and resources available for homeschool parents, teachers and students subscribe to Homeschool Success News -
http://hsnews.homeschooltutorpro.com
If you need an online tutor, courseware or wish to offer your services as a paid online tutor contact our TutorBug website: http://homeschooltutorpro.com
Magic Learning Systems also provides excellent products to enhance the homeschool experience for teachers and students. http://magic.homeschooltutorpro.com
Distance Learning – The Right Move for You?
By Edith Wright
Electronic education (e-learning), which today is analogous to Distance learning, has changed the way the world perceives education. Now, as never before, more and more people; who would otherwise not be able to further their prospects in the professional world through traditional education, may simply complete virtually any degree, course, certification or diploma, relative to nearly any field of study, at their own convenience and without ever having to step onto a campus at all.
Complete your basic education from an online high school or become an expert in your field of work with an online degree or PhD. Learn to use important software, or persue your lifelong passion. Anything you require, to further your employment prospects, get you a big raise at work or enrich you life in meaningful ways can now be achieved with minimal or no disruption to your daily life.
Distance learning from accredited online schools and colleges is considered to be education of a quality easily comparable to, and in many ways, better than traditional campus based learning. The trend has risen such that the number of people opting for distance learning courses is rising by 5 percent every year for nearly a decade, and by the fall of 2005, 96 percent of the colleges and universities in the US were offering online courses (amongst these are some of the best regarded traditional institutions like MIT and Yale) with some 3.2 million students benefiting (1). In 1999 a study concluded that, ‘Distance education is just as effective as traditional education in regards to learner outcomes’ (2) and in 2001, 56 percent of all domestic traditional learning institutions offered distance learning programs. An additional 12 percent of schools stated they planned on adding distance learning programs to their curriculum within the next few years (3).
This rising trend is evidence of the fact that today’s employers hold an optimistic-realistic and analytical view of distance learning, and what’s more? There is usually no mention upon a certificate, diploma or degree award as to how it was completed. An online degree, diploma or certificate is what it is – just as good as if you’d been studying on campus but without the inconvenience of having to work and live around lecture schedules, spending the additional time needed for commuting and the usual additional expenses inevitable to a traditional education experience. Just like in a traditional learning institution, you will have due dates for assignments, projects, papers, and tests and you would spend the same amount of time on homework, however, the added components of self-motivation and discipline are even more important.
Attractive as it may be, there is plenty of information prospective student must gain before they decide whether distance learning is right for them or before they choose an online college or course. We at schoolsgalore.com encourage prospective students/ trainees to thoroughly satisfy any doubts that they may have about the quality or validity of education provided by online universities colleges or schools that they explore.
Please find our list of accredited online schools, colleges and universities and our comprehensive A-Z guide to the various fields of study and the accredited courses offered online (which are accredited by either the U.S. Department of Education (USDE) or The Council for Higher Education Accreditation (CHEA) or both: unless explicitly mentioned otherwise).
Electronic education (e-learning), which today is analogous to Distance learning, has changed the way the world perceives education. Now, as never before, more and more people; who would otherwise not be able to further their prospects in the professional world through traditional education, may simply complete virtually any degree, course, certification or diploma, relative to nearly any field of study, at their own convenience and without ever having to step onto a campus at all.
Complete your basic education from an online high school or become an expert in your field of work with an online degree or PhD. Learn to use important software, or persue your lifelong passion. Anything you require, to further your employment prospects, get you a big raise at work or enrich you life in meaningful ways can now be achieved with minimal or no disruption to your daily life.
Distance learning from accredited online schools and colleges is considered to be education of a quality easily comparable to, and in many ways, better than traditional campus based learning. The trend has risen such that the number of people opting for distance learning courses is rising by 5 percent every year for nearly a decade, and by the fall of 2005, 96 percent of the colleges and universities in the US were offering online courses (amongst these are some of the best regarded traditional institutions like MIT and Yale) with some 3.2 million students benefiting (1). In 1999 a study concluded that, ‘Distance education is just as effective as traditional education in regards to learner outcomes’ (2) and in 2001, 56 percent of all domestic traditional learning institutions offered distance learning programs. An additional 12 percent of schools stated they planned on adding distance learning programs to their curriculum within the next few years (3).
This rising trend is evidence of the fact that today’s employers hold an optimistic-realistic and analytical view of distance learning, and what’s more? There is usually no mention upon a certificate, diploma or degree award as to how it was completed. An online degree, diploma or certificate is what it is – just as good as if you’d been studying on campus but without the inconvenience of having to work and live around lecture schedules, spending the additional time needed for commuting and the usual additional expenses inevitable to a traditional education experience. Just like in a traditional learning institution, you will have due dates for assignments, projects, papers, and tests and you would spend the same amount of time on homework, however, the added components of self-motivation and discipline are even more important.
Attractive as it may be, there is plenty of information prospective student must gain before they decide whether distance learning is right for them or before they choose an online college or course. We at schoolsgalore.com encourage prospective students/ trainees to thoroughly satisfy any doubts that they may have about the quality or validity of education provided by online universities colleges or schools that they explore.
Please find our list of accredited online schools, colleges and universities and our comprehensive A-Z guide to the various fields of study and the accredited courses offered online (which are accredited by either the U.S. Department of Education (USDE) or The Council for Higher Education Accreditation (CHEA) or both: unless explicitly mentioned otherwise).
Some Homeschooling Home-What you need to know
By Rebecca Walker
Homeschooling has come a long way from relative obscurity two decades ago. It is now a leading education trend in America as well as around the world. Estimates put the growth in the sector to about 15 percent per year. This has not stopped the spread of myths about homeschooling among many people; how many times do we often hear that homeschooled children are more likely to end up as social misfits, that they are unlikely to make it to college, that homeschooling will end up gobbling too much of your resources-both money and time, and so on?
Many research findings carried out suggest that not only do home schooled children have better social skills, but that they are less dependent on heir peers. Dr. Raymond Moore who wrote the book Better late than Early, found out that homeschooled individuals (those that have already become adults) are much more aware of the issues affecting their communities than the conventionally educated ones. They take their civil obligations like voting more seriously and many more belong to community organizations.
Experts believe that homeschooling contributes adequately to ideal child development, and since they learn their graces from many different age groups, not least their parents, they are capable of developing healthy relationships with just about anyone. Children who go to a traditional school on the other hand, get to spend time with their peers and not all of their time is spent learning in a classroom.
The first generation of homeschooled adults has already finished college and entered the job market. There are as many as 1,400 colleges that take in homeschooled students, so those facts should debunk the myth that homeschooled children are less likely to go to college. And the benefits of home school learning can be seen later in life as research suggests that those that took home learning have develop strong moral bases, exhibit greater work ethic and develop greater leadership skills as well.
There are fears that teaching children is a skill that not everyone has, and a parent would have hard time teaching subjects such as algebra. Well, it is undeniable that guiding your children how to read is not a simple task. But it’s a skill that any parent can learn, and especially motivated by the great benefits that comes with interacting with your children, like developing a strong bond. Secondly point is that one does not have to teach slightly technical subjects like algebra to children less than 10 years. This can wait until the child has matured a bit, and has expressed interest in science and math. Then you can look for ways to incorporate algebra into child leaning program, even using a tutor if you are not comfortable teaching the subject on your own.
You can also find a ways to home school the children without breaking the bank as teaching children at home can be expensive if you allow it to be. It is safe to assume that home schooling at home will be more costly than public schooling but less costly than private schooling. If you use a box curriculum or hook up with an independent study program, you can expect to pay a little more. There is an option of incorporating some public means to bring down costs, like using a public library, or public playground.
Homeschooling has come a long way from relative obscurity two decades ago. It is now a leading education trend in America as well as around the world. Estimates put the growth in the sector to about 15 percent per year. This has not stopped the spread of myths about homeschooling among many people; how many times do we often hear that homeschooled children are more likely to end up as social misfits, that they are unlikely to make it to college, that homeschooling will end up gobbling too much of your resources-both money and time, and so on?
Many research findings carried out suggest that not only do home schooled children have better social skills, but that they are less dependent on heir peers. Dr. Raymond Moore who wrote the book Better late than Early, found out that homeschooled individuals (those that have already become adults) are much more aware of the issues affecting their communities than the conventionally educated ones. They take their civil obligations like voting more seriously and many more belong to community organizations.
Experts believe that homeschooling contributes adequately to ideal child development, and since they learn their graces from many different age groups, not least their parents, they are capable of developing healthy relationships with just about anyone. Children who go to a traditional school on the other hand, get to spend time with their peers and not all of their time is spent learning in a classroom.
The first generation of homeschooled adults has already finished college and entered the job market. There are as many as 1,400 colleges that take in homeschooled students, so those facts should debunk the myth that homeschooled children are less likely to go to college. And the benefits of home school learning can be seen later in life as research suggests that those that took home learning have develop strong moral bases, exhibit greater work ethic and develop greater leadership skills as well.
There are fears that teaching children is a skill that not everyone has, and a parent would have hard time teaching subjects such as algebra. Well, it is undeniable that guiding your children how to read is not a simple task. But it’s a skill that any parent can learn, and especially motivated by the great benefits that comes with interacting with your children, like developing a strong bond. Secondly point is that one does not have to teach slightly technical subjects like algebra to children less than 10 years. This can wait until the child has matured a bit, and has expressed interest in science and math. Then you can look for ways to incorporate algebra into child leaning program, even using a tutor if you are not comfortable teaching the subject on your own.
You can also find a ways to home school the children without breaking the bank as teaching children at home can be expensive if you allow it to be. It is safe to assume that home schooling at home will be more costly than public schooling but less costly than private schooling. If you use a box curriculum or hook up with an independent study program, you can expect to pay a little more. There is an option of incorporating some public means to bring down costs, like using a public library, or public playground.
The Educational Advantages of Home schooling Students
By Anne Harvester
The advantages of a home schooled child over one who attends public school is quite noticeable. The home schooled student sees much more benefits and advantages than the student going to public school in so many ways. From scoring higher on standardized testing, to more one-on-one learning time, a curriculum tailored to their specific needs and abilities, in learning metric conversion or a metric calculator for example, is quite beneficial to a student craving learning.
It's easy to see why so many parents are choosing to home school their children. Once the decision is made to home school your child, what do you next and where do you start? Here are a few pieces of advice on how to get started in the right direction towards home school success:
Check Your State Requirements – The very first thing you should do is check the legal requirements of your particular state. While home schooling is legal in every state, the laws and requirements differ from one state to another. Some states require a letter of intent, quarterly reports and annual testing.
Choose a Curriculum – Get some research done as to the types of curriculum which work best for both you and your child. Math, History, Science, and English are common subjects taught throughout the United States. When choosing a one consider if you would prefer a more independent study or more instructional study? Are you comfortable teaching certain subjects more than others? For example, do you think you can better explain the metric conversion calculator or the metric table better than you can Civil War history?
Find the Proper Resources and Support – Do you know other parents who home school their own children? They can be a wonderful resource for advice and suggestions with any questions you may have. Look for a local home school support groups in your area or online for guidance. These support groups hold regular educational trips and activities such as to the zoo, museums and libraries that you can join in on.
Prepare The Home/School Environment – More than likely, your home will need to be rearranged to properly create a home schooling environment. True, most children don’t need a traditional desk and black board environment but they do require a designated area of the home to study all those metric table and metric conversion calculator notes you give them. The area must be organized and distraction-free to do assignments.
Set up Year End Goals – Your final step in the home school decision is to set up and plan specific goals for your child. At the end of the year, what would you have liked to achieve? It may not go the way you planned but it is a smart move to have a plan of action to navigate your way throughout the school year.
The advantages of a home schooled child over one who attends public school is quite noticeable. The home schooled student sees much more benefits and advantages than the student going to public school in so many ways. From scoring higher on standardized testing, to more one-on-one learning time, a curriculum tailored to their specific needs and abilities, in learning metric conversion or a metric calculator for example, is quite beneficial to a student craving learning.
It's easy to see why so many parents are choosing to home school their children. Once the decision is made to home school your child, what do you next and where do you start? Here are a few pieces of advice on how to get started in the right direction towards home school success:
Check Your State Requirements – The very first thing you should do is check the legal requirements of your particular state. While home schooling is legal in every state, the laws and requirements differ from one state to another. Some states require a letter of intent, quarterly reports and annual testing.
Choose a Curriculum – Get some research done as to the types of curriculum which work best for both you and your child. Math, History, Science, and English are common subjects taught throughout the United States. When choosing a one consider if you would prefer a more independent study or more instructional study? Are you comfortable teaching certain subjects more than others? For example, do you think you can better explain the metric conversion calculator or the metric table better than you can Civil War history?
Find the Proper Resources and Support – Do you know other parents who home school their own children? They can be a wonderful resource for advice and suggestions with any questions you may have. Look for a local home school support groups in your area or online for guidance. These support groups hold regular educational trips and activities such as to the zoo, museums and libraries that you can join in on.
Prepare The Home/School Environment – More than likely, your home will need to be rearranged to properly create a home schooling environment. True, most children don’t need a traditional desk and black board environment but they do require a designated area of the home to study all those metric table and metric conversion calculator notes you give them. The area must be organized and distraction-free to do assignments.
Set up Year End Goals – Your final step in the home school decision is to set up and plan specific goals for your child. At the end of the year, what would you have liked to achieve? It may not go the way you planned but it is a smart move to have a plan of action to navigate your way throughout the school year.
Compatible Numbers
By Chandrajeet K
• The word "compatible" means "well-matched". Compatible numbers are numbers that are friendly with each other.
For example:
15 and 5 are compatible numbers, because, 5 goes into 15 evenly ---- 15/5=3
So are 15 and 3, because, 3 goes into 15 evenly ---- 15/3=5
• Compatible numbers make estimation or mental calculation easier. They are useful in estimating the sum, difference, product, or quotient. Compatible numbers are close in value to the actual numbers given in the problem. They often end in 0 or 5.
Let’s look at a few examples.
Example 1
Estimate 33 + 28.
Look for close numbers that are easier to work with. Multiples of 10 are easier to work with.
28 is closer to 30.
Adding 33 and 30 is easy.
So,
33 + 28 is approximately equal to 33 + 30 = 63.
Example 2
Estimate 12 + 59 + 38.
Now, look for numbers that can be added together to make a 10 or multiples of 10.
12 + 59 + 38
= 12 + 38 + 59 ----- Swap the positions of 59 and 38
= 50 + 59 ------------ The 2 and the 8 are compatible. They add to make a 10.
= 109
Example 3
Estimate 52 – 37.
Look for close numbers that are easier to work with. Multiples of 10 are easier to work with.
37 is closer to 40.
Subtracting 40 from 52 is easy.
So,
52 – 37 is approximately equal to 52 – 40 = 12.
Example 4
Estimate the value of 61 × 5.8.
Compatible numbers for 61 and 5.8 are 60 and 6 respectively.
So, 61 × 5.8 is approximately equal to 60 × 6 = 360.
Example 5
Estimate 33/8.
To get a good estimate, round the dividend to the nearest multiple of the divisor.
Look for a number ‘close to 33 and at the same time is divisible by 8’. In other words, try to find a multiple of 8 that is close to 33.
32 is the right choice.
So,
33/8 is approximately equal to 32/8 = 4.
Example 6
Estimate 29/6.5.
29 and 6.5 are not friendly with each other. Try to find a pair of compatible numbers one of which is close to 29 and the other is near 6.5.
Round the numbers 29 and 6.5.
30, 6 is the ideal pair of compatible numbers with 30 close to 29 and 6 close to 6.5.
Therefore, 29 ÷ 6.5 is approximately equal to 30 ÷ 6 = 5.
Our estimate is 5. The actual quotient is 4.46. We are not for away from the actual answer.
Sometimes we should be careful about our choice of compatible numbers. It is important to choose compatible numbers that are appropriate to a given situation.
Let’s look at an example.
83 apples have to be packed in boxes. Each box holds 10 apples. About how many boxes will you need?
In order to find the number of boxes needed to pack ALL the apples, we have to divide 83 by 10. But…83 and 10 do not go well together.
Can you think of a number that goes well with 10 and at the same time is closer to 83?
Let’s try 80.
Well…80 is indeed friendly with 10, since 10 goes into 80 evenly.
But REMEMBER! We have to pack ALL 83 apples. 83 is greater than 80. If we consider 80, then we’ll have 3 apples left unpacked.
Think of another number…90?
Hooray!
90 is close to 83 and is friendly with 10 since 90 is a multiple of 10.
83/10 is approximately equal to 90/10 = 9.
Therefore you’ll need about 9 boxes.
• The word "compatible" means "well-matched". Compatible numbers are numbers that are friendly with each other.
For example:
15 and 5 are compatible numbers, because, 5 goes into 15 evenly ---- 15/5=3
So are 15 and 3, because, 3 goes into 15 evenly ---- 15/3=5
• Compatible numbers make estimation or mental calculation easier. They are useful in estimating the sum, difference, product, or quotient. Compatible numbers are close in value to the actual numbers given in the problem. They often end in 0 or 5.
Let’s look at a few examples.
Example 1
Estimate 33 + 28.
Look for close numbers that are easier to work with. Multiples of 10 are easier to work with.
28 is closer to 30.
Adding 33 and 30 is easy.
So,
33 + 28 is approximately equal to 33 + 30 = 63.
Example 2
Estimate 12 + 59 + 38.
Now, look for numbers that can be added together to make a 10 or multiples of 10.
12 + 59 + 38
= 12 + 38 + 59 ----- Swap the positions of 59 and 38
= 50 + 59 ------------ The 2 and the 8 are compatible. They add to make a 10.
= 109
Example 3
Estimate 52 – 37.
Look for close numbers that are easier to work with. Multiples of 10 are easier to work with.
37 is closer to 40.
Subtracting 40 from 52 is easy.
So,
52 – 37 is approximately equal to 52 – 40 = 12.
Example 4
Estimate the value of 61 × 5.8.
Compatible numbers for 61 and 5.8 are 60 and 6 respectively.
So, 61 × 5.8 is approximately equal to 60 × 6 = 360.
Example 5
Estimate 33/8.
To get a good estimate, round the dividend to the nearest multiple of the divisor.
Look for a number ‘close to 33 and at the same time is divisible by 8’. In other words, try to find a multiple of 8 that is close to 33.
32 is the right choice.
So,
33/8 is approximately equal to 32/8 = 4.
Example 6
Estimate 29/6.5.
29 and 6.5 are not friendly with each other. Try to find a pair of compatible numbers one of which is close to 29 and the other is near 6.5.
Round the numbers 29 and 6.5.
30, 6 is the ideal pair of compatible numbers with 30 close to 29 and 6 close to 6.5.
Therefore, 29 ÷ 6.5 is approximately equal to 30 ÷ 6 = 5.
Our estimate is 5. The actual quotient is 4.46. We are not for away from the actual answer.
Sometimes we should be careful about our choice of compatible numbers. It is important to choose compatible numbers that are appropriate to a given situation.
Let’s look at an example.
83 apples have to be packed in boxes. Each box holds 10 apples. About how many boxes will you need?
In order to find the number of boxes needed to pack ALL the apples, we have to divide 83 by 10. But…83 and 10 do not go well together.
Can you think of a number that goes well with 10 and at the same time is closer to 83?
Let’s try 80.
Well…80 is indeed friendly with 10, since 10 goes into 80 evenly.
But REMEMBER! We have to pack ALL 83 apples. 83 is greater than 80. If we consider 80, then we’ll have 3 apples left unpacked.
Think of another number…90?
Hooray!
90 is close to 83 and is friendly with 10 since 90 is a multiple of 10.
83/10 is approximately equal to 90/10 = 9.
Therefore you’ll need about 9 boxes.
Amicable Numbers
By Chandrajeet K
The term “amicable” means “friendly”.
Amicable numbers are friendly numbers.
Amicable numbers are a pair of numbers such that the sum of the proper divisors of each number equals the other number in the pair.
Okay…
What are proper divisors?
Proper divisors are all the numbers that divide evenly into a number, including 1 but excluding the number itself.
Example
1, 2, 3, 4, 6, and, 12 are divisors of the number 12. They divide evenly into 12 without a reminder.
But…the PROPER divisors of 12 are: 1, 2, 3, 4, and 6; we should NOT include 12 here.
Let’s go back to “Amicable Numbers”.
220 and 284 are the smallest pair of amicable numbers.
The proper divisors of 220 are:
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110
The sum of all these proper divisors is:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
Wow! That’s the other number in the pair.
Let’s now find the proper divisors of 284.
1, 2, 4, 71, and 142
Add them up.
1 + 2 + 4 + 71 + 142 = 220
Excellent! Indeed they are friendly numbers!
Easier said than done… It’s a lot of work to find divisors of big numbers.
Here are a few tips to help you finding the divisors and sum of the divisors of a number.
# 1
The quickest way to find the divisors of a number could be to obtain the prime factors of the number and then combine those factors in all possible ways.
Let’s look at an easy example.
You can use a factor tree to find the prime factors.
The prime factors of 24 are: 2, 2, 2, and 3
Let’s combine the factors in all possible ways to find the divisors of 24.
There are three 2’s and one 3 in 24.
Let’s first combine the 2’s.
2 is divisor of 24
2 x 2 = 4 is a divisor of 24
2 x 2 x 2 = 8 is a divisor of 24
Let’s now combine the 2’s and the 3.
3 is a divisor of 24
2 x 3 = 6 is a divisor of 24
2 x 2 x 3 = 12 is a divisor of 24
2 x 2 x 2 x 3 = 24 is a divisor of 24
So, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
We include 1 among the divisors, because ‘1 is a divisor of every number’.
Leaving out the 24 we get the PROPER divisors of 24.
The next step is to find the sum of proper factors of 24 which is:
1 + 2 + 3 + 4 + 6 + 8 + 12 = 36
Phew, that’s tedious! There’s got to be a better way.
What if we can get the SUM OF THE PROPER DIVISORS straight away?
Well…there’s a formula available for the sum of the divisors. This might come in handy when you got to work with pairs of really big amicable numbers.
# 2
Let ‘n’ be a positive integer.
n = (p^a) x (q^b) x (r^c) x … represents the prime factorization of ‘n’.
Then the sum of ALL the divisors of ‘n’ is:
{[p^(a+1)] – 1}/(p – 1) x {[q^(b+1)] – 1}/(q – 1) x {[r^(c+1)] – 1}/(r – 1) x …
Let’s try using this formula with the SAME SIMPLE example (24).
The prime factors of 24 are: 2, 2, 2, and 3
The prime factorization of 24 is (2^3) x (3^1).
So, the sum of ALL the divisors of 24 is:
{[2^ (3+1)] – 1}/(2 – 1) x {[3^(1+1)] – 1}/(3 – 1)
= {[2^4] – 1}/(1) x {[3^2] – 1}/(2)
= {[16] – 1}/(1) x {[9] – 1}/(2)
= {15}/(1) x {8}/(2)
= 15 x 4
= 60
REMEMBER 60 is the sum of ALL the divisors of 24. But what we need is the sum of the PROPER divisors of 24. So, SUBTRACT 24 from 60, we get:
60 – 24 = 36
Hooray! It works! It matches with our answer!
Hope this helps!
The term “amicable” means “friendly”.
Amicable numbers are friendly numbers.
Amicable numbers are a pair of numbers such that the sum of the proper divisors of each number equals the other number in the pair.
Okay…
What are proper divisors?
Proper divisors are all the numbers that divide evenly into a number, including 1 but excluding the number itself.
Example
1, 2, 3, 4, 6, and, 12 are divisors of the number 12. They divide evenly into 12 without a reminder.
But…the PROPER divisors of 12 are: 1, 2, 3, 4, and 6; we should NOT include 12 here.
Let’s go back to “Amicable Numbers”.
220 and 284 are the smallest pair of amicable numbers.
The proper divisors of 220 are:
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110
The sum of all these proper divisors is:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
Wow! That’s the other number in the pair.
Let’s now find the proper divisors of 284.
1, 2, 4, 71, and 142
Add them up.
1 + 2 + 4 + 71 + 142 = 220
Excellent! Indeed they are friendly numbers!
Easier said than done… It’s a lot of work to find divisors of big numbers.
Here are a few tips to help you finding the divisors and sum of the divisors of a number.
# 1
The quickest way to find the divisors of a number could be to obtain the prime factors of the number and then combine those factors in all possible ways.
Let’s look at an easy example.
You can use a factor tree to find the prime factors.
The prime factors of 24 are: 2, 2, 2, and 3
Let’s combine the factors in all possible ways to find the divisors of 24.
There are three 2’s and one 3 in 24.
Let’s first combine the 2’s.
2 is divisor of 24
2 x 2 = 4 is a divisor of 24
2 x 2 x 2 = 8 is a divisor of 24
Let’s now combine the 2’s and the 3.
3 is a divisor of 24
2 x 3 = 6 is a divisor of 24
2 x 2 x 3 = 12 is a divisor of 24
2 x 2 x 2 x 3 = 24 is a divisor of 24
So, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
We include 1 among the divisors, because ‘1 is a divisor of every number’.
Leaving out the 24 we get the PROPER divisors of 24.
The next step is to find the sum of proper factors of 24 which is:
1 + 2 + 3 + 4 + 6 + 8 + 12 = 36
Phew, that’s tedious! There’s got to be a better way.
What if we can get the SUM OF THE PROPER DIVISORS straight away?
Well…there’s a formula available for the sum of the divisors. This might come in handy when you got to work with pairs of really big amicable numbers.
# 2
Let ‘n’ be a positive integer.
n = (p^a) x (q^b) x (r^c) x … represents the prime factorization of ‘n’.
Then the sum of ALL the divisors of ‘n’ is:
{[p^(a+1)] – 1}/(p – 1) x {[q^(b+1)] – 1}/(q – 1) x {[r^(c+1)] – 1}/(r – 1) x …
Let’s try using this formula with the SAME SIMPLE example (24).
The prime factors of 24 are: 2, 2, 2, and 3
The prime factorization of 24 is (2^3) x (3^1).
So, the sum of ALL the divisors of 24 is:
{[2^ (3+1)] – 1}/(2 – 1) x {[3^(1+1)] – 1}/(3 – 1)
= {[2^4] – 1}/(1) x {[3^2] – 1}/(2)
= {[16] – 1}/(1) x {[9] – 1}/(2)
= {15}/(1) x {8}/(2)
= 15 x 4
= 60
REMEMBER 60 is the sum of ALL the divisors of 24. But what we need is the sum of the PROPER divisors of 24. So, SUBTRACT 24 from 60, we get:
60 – 24 = 36
Hooray! It works! It matches with our answer!
Hope this helps!
How to Choose the Best Institution to Pursue an Online MBA Program
By Glorence Lorence
Many working professionals dream of pursuing their educational aspirations that they could not fulfill earlier due to various reasons. Adding to the brunt is the current downturn in the market that has fuelled the need to excel and gain a competitive edge compared to other counterparts. In addition, today’s economic scenario is so bad that professionals cannot afford to quit jobs and pursue a certification or degree of their choice.
The ideal solution here is to opt for online courses. Online degree programs are flexible and convenient when compared to the traditional classroom teaching. Moreover, online courses and degree programs are widely accepted in the job market too. Therefore, anyone can pursue a degree without having to give up his or her current career and paycheck, of course.
Online courses are available for all degree programs as well as certification programs. All bachelors, masters and specialization courses are offered through online medium. The most popular among them is the online MBA degree because owning an MBA degree is a dream for any professional.
Top online MBA programs are offered by many leading universities and colleges. The following steps need to be followed while searching for a good institution.
1.Internet searches will provide many places that offer MBA programs.
2.Shortlist the institutions based on their proven track records
3.Send request for more information on course duration, fees and other details needed
4.Based on the information received, review fee structure, semester duration, project details etc and contact the course administrator for more details.
5.Be aware of ‘fake’ online programs and courses. Perform reference checks on the institution and with others who’ve already completed the course from that institution.
6.Make doubly sure how secure the mode of online communication will be.
7.Analyze the teaching methodology offered – e.g. Students with arts background who take up finance or marketing majors may have to take some filler courses.
After analyzing these points, choose the best institution for an online MBA program and kick start your dream course with confidence.
Many working professionals dream of pursuing their educational aspirations that they could not fulfill earlier due to various reasons. Adding to the brunt is the current downturn in the market that has fuelled the need to excel and gain a competitive edge compared to other counterparts. In addition, today’s economic scenario is so bad that professionals cannot afford to quit jobs and pursue a certification or degree of their choice.
The ideal solution here is to opt for online courses. Online degree programs are flexible and convenient when compared to the traditional classroom teaching. Moreover, online courses and degree programs are widely accepted in the job market too. Therefore, anyone can pursue a degree without having to give up his or her current career and paycheck, of course.
Online courses are available for all degree programs as well as certification programs. All bachelors, masters and specialization courses are offered through online medium. The most popular among them is the online MBA degree because owning an MBA degree is a dream for any professional.
Top online MBA programs are offered by many leading universities and colleges. The following steps need to be followed while searching for a good institution.
1.Internet searches will provide many places that offer MBA programs.
2.Shortlist the institutions based on their proven track records
3.Send request for more information on course duration, fees and other details needed
4.Based on the information received, review fee structure, semester duration, project details etc and contact the course administrator for more details.
5.Be aware of ‘fake’ online programs and courses. Perform reference checks on the institution and with others who’ve already completed the course from that institution.
6.Make doubly sure how secure the mode of online communication will be.
7.Analyze the teaching methodology offered – e.g. Students with arts background who take up finance or marketing majors may have to take some filler courses.
After analyzing these points, choose the best institution for an online MBA program and kick start your dream course with confidence.
The Counting Principle
By Chandrajeet K
The counting principle works for two or more events occurring in series. It is used to find the total number of possible outcomes when two or more events occur together.
The counting principle states that:
If a first event has ‘p’ possible outcomes followed by a second event that has ‘q’ possible outcomes followed by a third event that has ‘r’ possible outcomes and so on…, then there are a total of ‘p × q × r × …’ possible outcomes for the series of events.
We simply multiply the number of possible outcomes of each of the events in the series to get the total number of outcomes for the series.
Here are some examples using the counting principle:
Example 1
Helen has 4 blouses and 3 skirts. How many different outfits consisting of one blouse and one skirt are possible?
Solution
Helen has:
4 blouses – B1, B2, B3, B4
3 skirts – S1, S2, S3
The different possible outfits consisting of one blouse and one skirt are:
1. B1, S1
2. B1, S2
3. B1, S3
4. B2, S1
5. B2, S2
6. B2, S3
7. B3, S1
8. B3, S2
9. B3, S3
10. B4, S1
11. B4, S2
12. B4, S3
So:
Helen can make 12 different outfits consisting of one blouse and one skirt.
This can easily be calculated using the counting principle as 4 × 3 = 12.
Example 2
A restaurant offers a choice of 2 salads, 4 main courses, and 3 desserts. Find the number of possible choices when you choose one item from each category.
Solution
You have a choice of 2 salads – S1, S2
You have a choice of 4 main courses – M1, M2, M3, M4
You have a choice of 3 desserts – D1, D2, D3
As in the previous example, our list looks something like this…
The different possible choices consisting of a salad, a main course, and a dessert are:
1. S1, M1, D1
2. S1, M1, D2
3. S1, M1, D3
4. S1, M2, D1
5. S1, M2, D2,
6. S1, M2, D3
7. …
8. …
9. …
I'm not going to list all of them! I'll let you write them down.
By the counting principle we know there are a total of 2 × 4 × 3 = 24 possible choices of a 3-course meal.
Example 3
A state’s license plates have 3 letters followed by 3 digits.
(a) How many different license plates are possible if letters and digits can be repeated?
(b) How many different license plates are possible if letters and digits cannot be repeated?
Solution
(a) How many different license plates are possible if letters and digits can be repeated?
There are 26 choices for letters (A to Z) and 10 choices for digits (0, 1, 2, 3 …9).
The license plates have 3 letters followed by 3 digits.
LETTER LETTER LETTER DIGIT DIGIT DIGIT
Using the Counting Principle:
26 × 26 × 26 × 10 × 10 × 10
= 17,576,000
So, 17,576,000 different license plates are possible if letters and digits can be repeated.
(b) How many different license plates are possible if letters and digits cannot be repeated?
There are 26 choices for the first letter but only 25 choices for the second, and 24 for the third.
For the digits, there are 10 choices for the first, but only 9 for the second, and 8 for the third.
SO:
Using the Counting Principle:
LETTER LETTER LETTER DIGIT DIGIT DIGIT
26 × 25 × 24 × 10 × 9 × 8
= 11,232,000
So, 11,232,000 different license plates are possible if letters and digits cannot be repeated.
Here is a question for you…
How could the state increase the total number of possible license plates if letters and digits cannot be repeated?
The counting principle works for two or more events occurring in series. It is used to find the total number of possible outcomes when two or more events occur together.
The counting principle states that:
If a first event has ‘p’ possible outcomes followed by a second event that has ‘q’ possible outcomes followed by a third event that has ‘r’ possible outcomes and so on…, then there are a total of ‘p × q × r × …’ possible outcomes for the series of events.
We simply multiply the number of possible outcomes of each of the events in the series to get the total number of outcomes for the series.
Here are some examples using the counting principle:
Example 1
Helen has 4 blouses and 3 skirts. How many different outfits consisting of one blouse and one skirt are possible?
Solution
Helen has:
4 blouses – B1, B2, B3, B4
3 skirts – S1, S2, S3
The different possible outfits consisting of one blouse and one skirt are:
1. B1, S1
2. B1, S2
3. B1, S3
4. B2, S1
5. B2, S2
6. B2, S3
7. B3, S1
8. B3, S2
9. B3, S3
10. B4, S1
11. B4, S2
12. B4, S3
So:
Helen can make 12 different outfits consisting of one blouse and one skirt.
This can easily be calculated using the counting principle as 4 × 3 = 12.
Example 2
A restaurant offers a choice of 2 salads, 4 main courses, and 3 desserts. Find the number of possible choices when you choose one item from each category.
Solution
You have a choice of 2 salads – S1, S2
You have a choice of 4 main courses – M1, M2, M3, M4
You have a choice of 3 desserts – D1, D2, D3
As in the previous example, our list looks something like this…
The different possible choices consisting of a salad, a main course, and a dessert are:
1. S1, M1, D1
2. S1, M1, D2
3. S1, M1, D3
4. S1, M2, D1
5. S1, M2, D2,
6. S1, M2, D3
7. …
8. …
9. …
I'm not going to list all of them! I'll let you write them down.
By the counting principle we know there are a total of 2 × 4 × 3 = 24 possible choices of a 3-course meal.
Example 3
A state’s license plates have 3 letters followed by 3 digits.
(a) How many different license plates are possible if letters and digits can be repeated?
(b) How many different license plates are possible if letters and digits cannot be repeated?
Solution
(a) How many different license plates are possible if letters and digits can be repeated?
There are 26 choices for letters (A to Z) and 10 choices for digits (0, 1, 2, 3 …9).
The license plates have 3 letters followed by 3 digits.
LETTER LETTER LETTER DIGIT DIGIT DIGIT
Using the Counting Principle:
26 × 26 × 26 × 10 × 10 × 10
= 17,576,000
So, 17,576,000 different license plates are possible if letters and digits can be repeated.
(b) How many different license plates are possible if letters and digits cannot be repeated?
There are 26 choices for the first letter but only 25 choices for the second, and 24 for the third.
For the digits, there are 10 choices for the first, but only 9 for the second, and 8 for the third.
SO:
Using the Counting Principle:
LETTER LETTER LETTER DIGIT DIGIT DIGIT
26 × 25 × 24 × 10 × 9 × 8
= 11,232,000
So, 11,232,000 different license plates are possible if letters and digits cannot be repeated.
Here is a question for you…
How could the state increase the total number of possible license plates if letters and digits cannot be repeated?
Indirect Proof
By Chandrajeet K
The concept of proof is an important part of mathematics. There are three basic types of proofs: direct proofs, indirect proofs, and proofs by contradiction.
In this article, let’s learn about Indirect Proof. Please take time and read it carefully till the end.
Indirect proof is a type of proof that begins by ASSUMING what is to be proved is FALSE. Then we try to prove that our ASSUMPTION is true. If our ASSUMPTION leads to a contradiction then the original statement which was assumed false must be true.
Let me explain more in detail.
Suppose you wish to prove ‘statement A’ is true using an indirect proof.
The first thing you do is:
You assume statement A is false…and assume statement A’ which is a contrary of statement A to be true.
Then using valid arguments, you arrive at a contradiction (denial or disagreement) to statement A’.
Thus demonstrating that statement A is true.
This concept will be clearer when you look at some examples.
Example 1
Sarah left her house at 9:30 AM and arrived at her aunt’s house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.
Solution
Suppose that the given statement is false. That is: ‘Sarah did NOT exceed the 55 mph speed limit.
She drove 80 miles at 55 mph.
At this speed, Sarah would need 80/55 (approximately) = 1 hour 27 minutes to reach her aunt’s place.
But as per the problem she drove from 9:30 AM to 10:30 AM … exactly an hour.
SO, she must have driven faster than 55 mph….a contradiction to our assumption that Sarah did NOT exceed the speed limit.
Therefore, Sarah exceeded the speed limit.
Example 2
Prove the following using an indirect proof.
For all integers ‘n’, if 3n + 1 is even, then ‘n’ is odd.
Solution
Suppose that the conclusion is false. That is: ‘n’ is NOT odd.
Assume the contrary is true. That is: ‘n’ is even.
Then the statement contrary of the given statement is:
“For all integers ‘n’, if 3n + 1 is even, then ‘n’ is EVEN”
Let’s try to prove it.
‘n’ is even means ‘n’ is a multiple of 2…that is: n = 2m for some integer ‘m’.
Then:
3n + 1 = 3(2m) + 1 = 6m + 1 --- Call it Equation (1)
Well…6m is even. So, 6m + 1 is odd.
Therefore, 3n + 1 is ODD…because 3n + 1 = 6m + 1 from Equation (1).
By assuming ‘n’ is even, we’ve shown that 3n + 1 is ODD which is a contradiction to our assumption.
Therefore:
If ‘n’ is odd then 3n + 1 is even. This is the contrapositive of the statement to be proved.
Since the contrapositive is true, it follows that the original statement “if 3n + 1 is even, then ‘n’ is odd” is true.
The next example is a classic problem where an Indirect Proof is used.
Example 3
Prove that square root of 2 or SQRT (2) is irrational using an indirect proof.
Solution
ASSUME that the given statement is false.
That is:
SQRT(2) is NOT irrational.
Assume the contrary to be true…that is…SQRT(2) is RATIONAL.
Let’s try to prove it.
A rational number is a real number that can be expressed as a quotient of two integers a/b, where b does not equal 0.
We’ve assumed SQRT(2) to be a rational number.
So:
SQRT (2) = a/b. This fraction a/b is in lowest terms - that is, a and b have no common factors.
Multiply each side by ‘b’ to get rid of the fraction.
b × SQRT(2) = a
Square both sides.
SQR (b) × 2 = SQR (a) which is the same as:
2 SQR (b) = SQR (a) --- call it Equation (2)
SQR (a) is even…because from Equation (2) above, we have, SQR(a) = 2 SQR(b)…a multiple of 2.
SQR(a) is even…implies…’a’ is even. Then, a = 2k for some integer ‘k’.
Substitute a = 2k in Equation (2). We get:
2 SQR (b) = SQR (a) --- Equation (2)
2 SQR (b) = SQR (2k)
2 SQR (b) = 4 SQR (k)
Cancel ‘2’ on either side. We have:
SQR (b) = 2 SQR (k)
The above equation shows that ‘SQR(b)’ is even…because SQR(b) = 2 SQR(k).
Again, SQR (b) is even implies ‘b’ is even.
If ‘a’ and ‘b’ is both even, then they will have a common factor…
Then…how can the fraction a/b be in lowest terms?
A contradiction…
SO, SQRT (2) is IRRATIONAL.
Example 4
Prove that “For all integers ‘n’, if ‘n’ is odd then SQR(n) is odd” using an indirect proof.
Solution
Suppose the conclusion is false.
That is:
SQR(n) is NOT odd.
ASSUME the contrary… SQR (n) is even.
Then the statement contrary of the given statement is:
“For all integers ‘n’, if ‘n’ is odd then SQR(n) is even”
Let’s try to prove it.
If SQR(n) is even, then SQR(n) can be expressed as a multiple of 4.
So:
SQR (n) = 4k for some integer ‘k’.
Take square root on either sides of the equation. We get:
n = 2 SQRT (k)
The above equation shows that ‘n’ is even, because ‘n’ is a multiple of 2…
By assuming ‘SQR(n)’ is even, we’ve shown that ‘n’ is EVEN which is a contradiction to our assumption.
So:
If ‘SQR(n)’ is odd then ‘n’ is odd. This is the contrapositive of the statement to be proved.
Since the contrapositive is true, it follows that the original statement “If ‘n’ is odd then ‘SQR (n)’ is odd” is true.
The concept of proof is an important part of mathematics. There are three basic types of proofs: direct proofs, indirect proofs, and proofs by contradiction.
In this article, let’s learn about Indirect Proof. Please take time and read it carefully till the end.
Indirect proof is a type of proof that begins by ASSUMING what is to be proved is FALSE. Then we try to prove that our ASSUMPTION is true. If our ASSUMPTION leads to a contradiction then the original statement which was assumed false must be true.
Let me explain more in detail.
Suppose you wish to prove ‘statement A’ is true using an indirect proof.
The first thing you do is:
You assume statement A is false…and assume statement A’ which is a contrary of statement A to be true.
Then using valid arguments, you arrive at a contradiction (denial or disagreement) to statement A’.
Thus demonstrating that statement A is true.
This concept will be clearer when you look at some examples.
Example 1
Sarah left her house at 9:30 AM and arrived at her aunt’s house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.
Solution
Suppose that the given statement is false. That is: ‘Sarah did NOT exceed the 55 mph speed limit.
She drove 80 miles at 55 mph.
At this speed, Sarah would need 80/55 (approximately) = 1 hour 27 minutes to reach her aunt’s place.
But as per the problem she drove from 9:30 AM to 10:30 AM … exactly an hour.
SO, she must have driven faster than 55 mph….a contradiction to our assumption that Sarah did NOT exceed the speed limit.
Therefore, Sarah exceeded the speed limit.
Example 2
Prove the following using an indirect proof.
For all integers ‘n’, if 3n + 1 is even, then ‘n’ is odd.
Solution
Suppose that the conclusion is false. That is: ‘n’ is NOT odd.
Assume the contrary is true. That is: ‘n’ is even.
Then the statement contrary of the given statement is:
“For all integers ‘n’, if 3n + 1 is even, then ‘n’ is EVEN”
Let’s try to prove it.
‘n’ is even means ‘n’ is a multiple of 2…that is: n = 2m for some integer ‘m’.
Then:
3n + 1 = 3(2m) + 1 = 6m + 1 --- Call it Equation (1)
Well…6m is even. So, 6m + 1 is odd.
Therefore, 3n + 1 is ODD…because 3n + 1 = 6m + 1 from Equation (1).
By assuming ‘n’ is even, we’ve shown that 3n + 1 is ODD which is a contradiction to our assumption.
Therefore:
If ‘n’ is odd then 3n + 1 is even. This is the contrapositive of the statement to be proved.
Since the contrapositive is true, it follows that the original statement “if 3n + 1 is even, then ‘n’ is odd” is true.
The next example is a classic problem where an Indirect Proof is used.
Example 3
Prove that square root of 2 or SQRT (2) is irrational using an indirect proof.
Solution
ASSUME that the given statement is false.
That is:
SQRT(2) is NOT irrational.
Assume the contrary to be true…that is…SQRT(2) is RATIONAL.
Let’s try to prove it.
A rational number is a real number that can be expressed as a quotient of two integers a/b, where b does not equal 0.
We’ve assumed SQRT(2) to be a rational number.
So:
SQRT (2) = a/b. This fraction a/b is in lowest terms - that is, a and b have no common factors.
Multiply each side by ‘b’ to get rid of the fraction.
b × SQRT(2) = a
Square both sides.
SQR (b) × 2 = SQR (a) which is the same as:
2 SQR (b) = SQR (a) --- call it Equation (2)
SQR (a) is even…because from Equation (2) above, we have, SQR(a) = 2 SQR(b)…a multiple of 2.
SQR(a) is even…implies…’a’ is even. Then, a = 2k for some integer ‘k’.
Substitute a = 2k in Equation (2). We get:
2 SQR (b) = SQR (a) --- Equation (2)
2 SQR (b) = SQR (2k)
2 SQR (b) = 4 SQR (k)
Cancel ‘2’ on either side. We have:
SQR (b) = 2 SQR (k)
The above equation shows that ‘SQR(b)’ is even…because SQR(b) = 2 SQR(k).
Again, SQR (b) is even implies ‘b’ is even.
If ‘a’ and ‘b’ is both even, then they will have a common factor…
Then…how can the fraction a/b be in lowest terms?
A contradiction…
SO, SQRT (2) is IRRATIONAL.
Example 4
Prove that “For all integers ‘n’, if ‘n’ is odd then SQR(n) is odd” using an indirect proof.
Solution
Suppose the conclusion is false.
That is:
SQR(n) is NOT odd.
ASSUME the contrary… SQR (n) is even.
Then the statement contrary of the given statement is:
“For all integers ‘n’, if ‘n’ is odd then SQR(n) is even”
Let’s try to prove it.
If SQR(n) is even, then SQR(n) can be expressed as a multiple of 4.
So:
SQR (n) = 4k for some integer ‘k’.
Take square root on either sides of the equation. We get:
n = 2 SQRT (k)
The above equation shows that ‘n’ is even, because ‘n’ is a multiple of 2…
By assuming ‘SQR(n)’ is even, we’ve shown that ‘n’ is EVEN which is a contradiction to our assumption.
So:
If ‘SQR(n)’ is odd then ‘n’ is odd. This is the contrapositive of the statement to be proved.
Since the contrapositive is true, it follows that the original statement “If ‘n’ is odd then ‘SQR (n)’ is odd” is true.
Homeschool Graduations Announcements for the Graduate
By sarah porter
Homeschool Graduations Announcements for the Graduate
In the last two decades, more and more young people have been getting their education at home. The home schooling movement does not just include people from strict religious backgrounds, although they did help gain acceptance for this approach to education. Today, millions of students are beginning to complete their school requirements and are preparing for homeschool graduations.
Graduate from Home School
If you or your child has been taught at home, you need to prepare for graduation. Different states have different requirements for graduation. However, most states take a very hands-off approach to this method. That means the parents or the overseer of the education determines graduation. Students must meet the minimum requirements for graduation established by the state. Good records should be maintained.
In some cities, you can find home school graduate programs that allow students to go through a ceremony similar to that experienced by traditional graduates. The ceremonies are usually small and are sometimes held at local churches. If you're part of a Homeschooling group, you can ask about these programs or you could even work together to form your own if a large number of students are nearing the end of their education.
Although standardized tests aren't going to be required for the homeschool graduate to complete his or her basic studies, the ACT and/or SAT will be required if he or she plans to attend college. The good news is that most colleges and universities today are welcoming students who attended school at home with open arms. That's because so many of them have been proven to be overachievers.
Mailing Announcements
Typically, you should start searching for graduation announcements prior to the completion of your child's education. Of course, if there's not going to be a specific ceremony, you can pick the home school stationery any time you desire. Most parents will wait until their child has completed the requirements, until the child has passed the ACT or SAT or until the child has been accepted to a college before mailing out the announcements. In these situations, the parents can help celebrate several pieces of good news for their child at the same time.
Remember to send the announcements to all of your friends and family members, even those people who did not necessarily agree with your decision to have homeschool graduations. If your family or child is active in church, in clubs or in other types of groups, be sure to send announcements to them as well. Leave off any type of gift registry details since presents are not required with any type of graduation. You don't want any one to feel obligated.
If you are going to have a graduation party, choose to mail out invitations instead of announcements. Be sure to include RSVP information, too. You may also want to send out invitations if you are going to participate in one of the home school graduation ceremonies. Keep in mind that some people may not give homeschool graduations as much respect as traditional graduations but the important thing is to remember that you've worked hard to get where you are and are now ready to tackle the next big education challenge.
Some Home School shoppes offer MANY advantages to shopping with them, the largest and best place to purchase online invitations and announcements cards, such as:
* Unique and Exclusive Designs Available
* Print and Ship YOUR order on the SAME DAY of Approval
* Add a photo, a picture or a logo to any card available on Their site
* Will Modify any card design or color available though Their site – just ask
* Will design exactly what you want for you
* Proofs are Emailed to You within ONE Hour during the Business Day, and You Can Make Unlimited Changes at No Additional Cost
* Free 10 Cards and Free Shipping with
* View Your Personalized Homeschool Graduations PRIOR to buying
Homeschool Graduations Announcements for the Graduate
In the last two decades, more and more young people have been getting their education at home. The home schooling movement does not just include people from strict religious backgrounds, although they did help gain acceptance for this approach to education. Today, millions of students are beginning to complete their school requirements and are preparing for homeschool graduations.
Graduate from Home School
If you or your child has been taught at home, you need to prepare for graduation. Different states have different requirements for graduation. However, most states take a very hands-off approach to this method. That means the parents or the overseer of the education determines graduation. Students must meet the minimum requirements for graduation established by the state. Good records should be maintained.
In some cities, you can find home school graduate programs that allow students to go through a ceremony similar to that experienced by traditional graduates. The ceremonies are usually small and are sometimes held at local churches. If you're part of a Homeschooling group, you can ask about these programs or you could even work together to form your own if a large number of students are nearing the end of their education.
Although standardized tests aren't going to be required for the homeschool graduate to complete his or her basic studies, the ACT and/or SAT will be required if he or she plans to attend college. The good news is that most colleges and universities today are welcoming students who attended school at home with open arms. That's because so many of them have been proven to be overachievers.
Mailing Announcements
Typically, you should start searching for graduation announcements prior to the completion of your child's education. Of course, if there's not going to be a specific ceremony, you can pick the home school stationery any time you desire. Most parents will wait until their child has completed the requirements, until the child has passed the ACT or SAT or until the child has been accepted to a college before mailing out the announcements. In these situations, the parents can help celebrate several pieces of good news for their child at the same time.
Remember to send the announcements to all of your friends and family members, even those people who did not necessarily agree with your decision to have homeschool graduations. If your family or child is active in church, in clubs or in other types of groups, be sure to send announcements to them as well. Leave off any type of gift registry details since presents are not required with any type of graduation. You don't want any one to feel obligated.
If you are going to have a graduation party, choose to mail out invitations instead of announcements. Be sure to include RSVP information, too. You may also want to send out invitations if you are going to participate in one of the home school graduation ceremonies. Keep in mind that some people may not give homeschool graduations as much respect as traditional graduations but the important thing is to remember that you've worked hard to get where you are and are now ready to tackle the next big education challenge.
Some Home School shoppes offer MANY advantages to shopping with them, the largest and best place to purchase online invitations and announcements cards, such as:
* Unique and Exclusive Designs Available
* Print and Ship YOUR order on the SAME DAY of Approval
* Add a photo, a picture or a logo to any card available on Their site
* Will Modify any card design or color available though Their site – just ask
* Will design exactly what you want for you
* Proofs are Emailed to You within ONE Hour during the Business Day, and You Can Make Unlimited Changes at No Additional Cost
* Free 10 Cards and Free Shipping with
* View Your Personalized Homeschool Graduations PRIOR to buying
Mathematics in Physics and Chemistry
By Chandrajeet K
Mathematics is the queen of all sciences’ – those are the words of Carl Friedrich Gauss the greatest mathematician of all time.
Mathematics is an important tool for science. Math is most widely used in other sciences. Physics, Chemistry, astronomy, engineering rely most heavily upon mathematical ideas.
Students who consider studying Physics or Chemistry will need a relatively strong Math background.
Mathematics in Physics
Physics is the natural science which explores concepts like mass, energy, matter and its motions. Strong foundation in Algebra, Trigonometry, Geometry, and calculus is essential for physics. Mathematical methods are absolutely necessary to deal with important concepts in physics.
The following are some examples.
(1) Electromagnetic theory is the branch of physics that studies the group of forces associated with electric charges. Vector Analysis is very important for the understanding and developing of Electromagnetic theory.
(2) Group theory is useful in Spectroscopy, Quantum mechanics, Solid state physics, and Nuclear physics.
(3) Fourier techniques are important for the analysis of all linear systems in physics.
(4) Matrix Analysis is necessary for understanding Quantum Mechanics.
(5) Complex numbers are used extensively in physics to describe Electromagnetic Waves and Quantum Mechanics.
Mathematics in Chemistry
Chemistry is the natural science which explores the composition and properties of substances. Math is essential for chemistry. The necessary mathematical background for the study of chemistry includes basic algebra, some trigonometry, and calculus.
The following are some examples.
(1) Being able to balance chemical equations is a very important skill for chemistry students. It’s a simple mathematical exercise. Balancing a chemical equation refers to establishing the mathematical relationship between the amounts of reactants and products involved in the chemical reaction.
Let’s go more in detail.
A chemical equation is a statement that describes what happens in a chemical reaction.
In a chemical equation, we place the reactants (substances undergoing chemical reaction) on the left side of the equation and the products (substances produced in a chemical reaction) on the right side of the equation. We have reactants and products separated by an arrow and the arrow always points in the direction of the products.
Consider the reaction of carbon with oxygen gas to produce carbon-dioxide.
C + O2 ---> CO2 (2 is subscript)
The above equation is already balanced, because, it has an equal number of atoms of each element in the reactants and the product. One carbon atom (C) and two oxygen atoms (O) on the left side of the equation and it’s the same on the right side too.
Let’s look at one more example.
Sodium chloride is the common salt. Sodium and chlorine form sodium chloride.
Na + Cl2 ---> NaCl (2 is subscript)
The above equation is NOT balanced. It has two chlorine atoms on the left side, but, only one on the right side of the equation.
Let’s balance this chemical equation.
2Na + Cl2 ---> 2NaCl (2 is subscript only in Cl2)
It works! Notice that now there are equal number of atoms of each element in the reactants and the product.
Chemical equations can be balanced conveniently using matrices or simultaneous equations.
A number of fields of chemistry use a significant amount of Math.
(2) Electrochemistry is a branch of chemistry that studies the chemical action of electricity and the production of electricity by chemical reactions. Diffusion in electrochemistry is completely based on differential equations.
(3) Biochemistry is the study of the chemical processes in living organisms. Even biochemistry has important topics which depend heavily on binding theory and kinetics.
Mathematics is the queen of all sciences’ – those are the words of Carl Friedrich Gauss the greatest mathematician of all time.
Mathematics is an important tool for science. Math is most widely used in other sciences. Physics, Chemistry, astronomy, engineering rely most heavily upon mathematical ideas.
Students who consider studying Physics or Chemistry will need a relatively strong Math background.
Mathematics in Physics
Physics is the natural science which explores concepts like mass, energy, matter and its motions. Strong foundation in Algebra, Trigonometry, Geometry, and calculus is essential for physics. Mathematical methods are absolutely necessary to deal with important concepts in physics.
The following are some examples.
(1) Electromagnetic theory is the branch of physics that studies the group of forces associated with electric charges. Vector Analysis is very important for the understanding and developing of Electromagnetic theory.
(2) Group theory is useful in Spectroscopy, Quantum mechanics, Solid state physics, and Nuclear physics.
(3) Fourier techniques are important for the analysis of all linear systems in physics.
(4) Matrix Analysis is necessary for understanding Quantum Mechanics.
(5) Complex numbers are used extensively in physics to describe Electromagnetic Waves and Quantum Mechanics.
Mathematics in Chemistry
Chemistry is the natural science which explores the composition and properties of substances. Math is essential for chemistry. The necessary mathematical background for the study of chemistry includes basic algebra, some trigonometry, and calculus.
The following are some examples.
(1) Being able to balance chemical equations is a very important skill for chemistry students. It’s a simple mathematical exercise. Balancing a chemical equation refers to establishing the mathematical relationship between the amounts of reactants and products involved in the chemical reaction.
Let’s go more in detail.
A chemical equation is a statement that describes what happens in a chemical reaction.
In a chemical equation, we place the reactants (substances undergoing chemical reaction) on the left side of the equation and the products (substances produced in a chemical reaction) on the right side of the equation. We have reactants and products separated by an arrow and the arrow always points in the direction of the products.
Consider the reaction of carbon with oxygen gas to produce carbon-dioxide.
C + O2 ---> CO2 (2 is subscript)
The above equation is already balanced, because, it has an equal number of atoms of each element in the reactants and the product. One carbon atom (C) and two oxygen atoms (O) on the left side of the equation and it’s the same on the right side too.
Let’s look at one more example.
Sodium chloride is the common salt. Sodium and chlorine form sodium chloride.
Na + Cl2 ---> NaCl (2 is subscript)
The above equation is NOT balanced. It has two chlorine atoms on the left side, but, only one on the right side of the equation.
Let’s balance this chemical equation.
2Na + Cl2 ---> 2NaCl (2 is subscript only in Cl2)
It works! Notice that now there are equal number of atoms of each element in the reactants and the product.
Chemical equations can be balanced conveniently using matrices or simultaneous equations.
A number of fields of chemistry use a significant amount of Math.
(2) Electrochemistry is a branch of chemistry that studies the chemical action of electricity and the production of electricity by chemical reactions. Diffusion in electrochemistry is completely based on differential equations.
(3) Biochemistry is the study of the chemical processes in living organisms. Even biochemistry has important topics which depend heavily on binding theory and kinetics.
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