Sum: Sum in common terms is a way of putting things together or getting a total. In maths, sum is the answer that we get on adding two or more numbers.
Symbol Used:
The symbol used to represent a sum is ‘+’, also known as plus.
Common Terms used:
Some common alternative terms that are used other than sum or depict a need to add to or more numbers are “total”, “together”, “all-together” and “in all”.
Representation:
a + b = c
Here, ‘a’ and ‘b’ are known as addends and ‘c’ is known as sum.
The terms or numbers that participate in the addition or sum are called addends. The minimum number of addends is two.
For example:
12 + 13 = 25;
In this mathematical expression, 12 and 13 are addends and 25 is the sum.
How to calculate a sum?
Sum can be calculated in two ways. Let’s discuss:
Problem: Find the sum of 8 + 3?
Way 1: Pictorial representation
Here you need to draw pictures/lines as per the number of addends and the count them together.
So, for 8 =
For 3 =
Now, count all the stars together. They come out to be 11.
So, the sum of 8 + 3 is 11.
This method turns out to be very tedious if large numbers are involved as drawing so many pictures would be very difficult. But, if you have just started learning about sum then, this is the ideal way to learn.
Way 2: Count on fingers
This approach makes use of your ten fingers to calculate the sum.
Here you need to keep the biggest addend in your mind which is 8 in this case. Now, open the fingers equal to the second added.
Here, our second addend is 3. So, open three fingers.
Now start counting ahead of the number in your mind and start closing the open fingers one by one.
Start from 9 (next number of 8) and close your first open finger.
10, close your second finger.
Finally, 11 close your third finger. Now, stop when all fingers are closed.
So, on adding 8 and 3, the sum comes out to be 11.
Now, Let’s practice another problem:
Find the sum of 11 + 5?
Way 1
11= | | | | | | | | | | |
5 = | | | | |
On counting all the pipes the sum comes out to be 16.
Way 2
11 in the mind and open 5 fingers
on counting ahead of 11 and simultaneously closing the five open fingers we get 16.
So, the sum of 11 and 5 is 16 or, 11 + 5 = 16.
Real-world application
We use sum to add individual item prices when we go to shopping and pay the sum to the shopkeeper.
Sum Trivia
Adding zero to any number will result in the same number. This is also called Additive Identity Property. For example : 19 + 0 = 19
On adding 1 to any number, the sum would be the successor of that number. Example: 6 + 1 =7
The sum remains the same irrespective of the order of addends. Example: 4 + 15 = 15 + 4
The sum is always greater than individual addends.
Range: Given a set of numbers, range is defined as the difference between the highest value and the lowest value. It is one of the methods used to find the central tendency in statistics. Some other similar methods are mean, median and mode that help us to give information and draw conclusions about a given data set.
Range helps us to understand the span or spread of data. Also, it helps to identify the gap between the largest and the smallest number.
How to find range? Formula
To find range, subtract the least value from the highest value from a given set of numbers.
Formula:
Range = Largest number on the set – smallest number on the set
Let’s practice with this problem:
Here’s a given set: 23, 56, 32, 89, 70, 45 and 36
Identify the smallest and largest number from the set. If you cannot do it orally, try to arrange the set in ascending order and then take the highest and lowest value.
Largest number = 89
Smallest number = 23
Range = 89 – 23 = 66
Mid Range
Mid range is the number between the lowest and highest value of a given set of numbers. Just add the two numbers and take their average or divide by two.
So, range and mid range of {23, 56, 32, 89, 70, 45, 36} is 66 and 56 respectively.
Let’s take a couple of more examples:
Example 1
Marks of students in class A
30, 40, 30, 45, 35, 43, 23, 44
Range :
45 – 23 = 22
Mid Range:
(45 +23) / 2 = 68 / 2 = 34
Example 2
Marks of students in class B
30, 40, 45, 40, 30, 40, 35, 40
Range :
45 – 30 = 15
Mid Range:
(45 +30) / 2 = 75 / 2 = 37.5
Observation:
As we can see that range of class B is smaller than class A, this means marks distribution in class B is more clustered (more close) in class B than class A.
Also, range is ineffective even if the middle values are changed but the largest and smallest remain the same. So, if your middle values are varying largely it’s better to opt other methods such as mean, median and mode.
Why do we use Range? Applications
Range offers a brilliant way to understand the spread of a given data set. It is very easy to calculate as it involves basic arithmetic operations. The range is also used in other mathematical formulas of statistics such as standard deviation.
Another application of Range can be seen in whiskers plot or boxplot. The highest and lowest values of a given set a plotted at the end of on a whiskers graph. Now, the total length of box and whiskers is equal to the range.
Product: The term product in math means adding a number repetitively till specified number of times. The operator used is ‘*’ or sometimes ‘x’. It is also known as multiply or multiplication. A product is an answer when two number are multiplied.
For example, check this mathematical statement:
a x b=c This statement is read as ‘a’ times ‘b’ equals to ‘c’
or, when we add ‘b’ repetitively ‘a’ time, the answer is ‘c’.
or ‘a’ multiplied by ‘b’ results in ‘c’.
or ‘a’ into ‘b’ is ‘c’.
or the product of ‘a’ and ‘b’ is ‘c’.
Here, ‘a’ and ‘b’ are known as multiplicand and multiplier respectively. ‘c’ is referred to as the product.
Let’s learn it by taking an example:
Example 1:
Say, we need to find product of 3 and 6.
There are two approaches to attain the same.
Approach 1: Direct multiplication
Here, 3 is multiplicand and 6 is the multiplier. Just skim through the table of 6 and now you know the answer
Its, 3 x 6 = 18 (3 times 6 is 18)
So, the product of 3 and 6 is 18.
Approach 2: Repetitive Addition
Here you need to add ‘6’ , three times.
6 + 6 + 6 = 18
The answer is 18.
Still not clear, Let’s take another example
Example 2:
Now we need to find the product of 8 and 9.
By following approach 1, you just read the table of 9 in your mind and find the answer.
So, 8 x 9 = 72 (8 times 9 is 72)
Hence, the product of 8 and 9 is 72.
Approach two or repetitive addition says you need to add 9, eight times.
So, 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 =72.
Or we can say that 8 times 9 is 72.
Example 3: Let’s discuss a Word problem
Suppose there are 10 chocolates in a box. Find total number of chocolates in 7 boxes?
Approach 1
First, just multiply number of chocolates in one box and total number of boxes.
Number of chocolates in one box = 10
number of chocolates in 7 boxes= 10 x 7=70
The product of 7 and 10 will give total number of chocolates i.e., 70.
Approach 2
Other way is repetitive addition.
We add the number of chocolates in each box.
10 + 10 + 10 +10 + 10 + 10 + 10 =70.
In both the scenarios, the answer comes out to be 70.
Let’s discuss some basic properties of product.
There are four basic properties of product. Let’s check them out.
Commutative Property
Regardless of the order, the product of two number will remain the same.
For example: 5 x1 7 = 1 7 x 5
Associative Property
When three numbers are multiplied, the product remains the same regardless of which two numbers are multiplied first.
13 X (2 X 6) = (13 X 2) X 6
Multiplicative Identity property
This property stated that when two number are multiplied and one of them is ‘1’, the product is the number itself.
Do you feel nervous while performing mathematical calculations? Do you also suffer from the fear of doing math? Then remember you are not alone! You may be suffering from math anxiety only! Here’s all you need to know about it!
TABLE OF CONTENT
Introduction
What is math anxiety?
Causes of math anxiety
Contributing factors
Quotes often said by an anxious student
Signs and symptoms
Coping with math anxiety
Additional tips
Bottom Line
Introduction
Math is not only about performing calculations and limiting students to textbook problems only. Maths is a way of life that is used at almost every phase of life.
Whether you go to the business world, choose a career option, or buy something, everything includes the math calculations. So, the students need to understand the answer to “WHY” behind what they are learning.
It is fair to say that math is not everyone’s cup of tea or say it is not everyone’s favourite subject. For many people, it is indeed a matter of anxiety and a feeling of tension that arises whenever they think about solving the equation.
What exactly will be the correct answer?
Will I be able to find the correct answer to the equation?
How to solve the equation?
Will it affect the grades?
Will I be able to pass the examination?
Will it affect my image in front of society?
And many more, these questions result in a child starting to avoid the things involving maths in it rather than choose to cope with it.
If this is the same case with you, let us dive into the topic to determine what measures can help your child shine in maths and overcome their weakness.
What is maths anxiety?
Mathematical anxiety, often known as math phobia, refers to one’s ability to do maths. Students often develop a sense of anxiety towards the math subject, and as a result, they end up avoiding the mathematical calculations.
Looking into the essentiality of addressing the problem, you will be happy to know that now the problem of math anxiety can be addressed by just following a few essential tips.
Yes, you heard it right! But before thinking of overcoming math anxiety, it is essential to get familiar with the causes of math anxiety so that you can quickly get the indication when your child needs help from you.
Causes of math anxiety
Fear of being wrong: The fear of being wrong or always being proper to maintain a strong image in front of the other students is also the basis of a child suffering from math anxiety.
Parents’ negative predispositions: Positive attitude towards the child can drastically change the student’s perceptions of doing something well. Consequently, the negative predispositions can actually break the whole scenario into a completely different zone.
The pressure of timed tests: The deadline of time limits imposed on students makes them feel anxious. This leads to forgetting concepts and only focusing on the grades. However, the tests can negatively impact the grades due to fear, so the failure is probably more. This indeed creates a vicious circle that is challenging to break.
Teacher influence: Students quickly pick up their teacher’s feelings about the subject. If the teacher has a positive attitude and stays excited while teaching maths, the same will be reflected in students.
While if the teacher is negative and least interested in explaining the concepts well, the same effect will be seen in students.
Unwillingness to complete assignments: The act of avoiding the mathematical calculations due to a mindset of disinterest in the subject or unwillingness to perform it can be a more significant cause of the anxiety.
The positive approach and genuine support can help the child to cope with the anxiety well.
Other factors that contribute to math anxiety
Bad grades at school
Lack of encouragement from teachers or parents
Gender stereotypes
Lack of positive role models
Math problems being used as a punishment in schools
Several quotes are often said by students suffering from math anxiety!
Some people can do math- not me!
Whenever I look at a math problem, my mind goes completely blank!
Math subjects and exams terrify me! My palms get clammy, I breathe fast and whatnot!
Math anxiety: Signs and symptoms
If your child seems struggling particularly with math concepts or grades, then do not forget to look for these math anxiety symptoms. Probably your child needs your help to cope up with the subject.
Negative Self-talk
Students experiencing math anxiety often witness negative self-talk as a common symptom. Is your child also doing the same?
Does your child use the statements like “I will never be good at maths”, or I am unable to solve mathematical equations, or I hate math? If yes, then probably your child needs help with the subject.
Unusual nervousness while working on mathematical equations or thinking about it.
Even thinking about the math subject can be enough reason to cause stress to the student. They often feel nervous while thinking about the subject or working on it.
Panic during tests
The traditional classroom environment becomes the primary stress source for a student, especially during a test or they are expected to contribute in the class.
Intense emotional reactions
Math anxiety manifests emotionally as well. In this case, students quickly start panicking, get teary-eyed, or grow angry during math class.
Permanency feeling
When math is not a specific student’s strong suit, they instantaneously believe that it is just the way it is and nothing can be done about it. And the moment they start believing this, they quickly give up and lower down their motivation.
Low achievement
Due to the lack of confidence, answering the questions appropriately during a test is never an option. And as their exposure to the subject decreases, it drastically affects the overall performance.
Avoidance
Ample students experiencing math anxiety choose to avoid the math lectures, tests, or quiz. The reason is that they try to find an escape from the subject, and so they start avoiding it. Indeed it has become a nightmare to them, which is why they often do this.
Physiological effects
The research studies have suggested that the students with math anxiety experience increased heart rate, nervousness, upset stomach, clammy hands, and lightheadedness.
How to overcome math anxiety?
Out of many students classified to be bad at math, many could be experiencing math anxiety which is undoubtedly a solvable problem.
Yes, you heard it right!
The maths fear is a real phenomenon among school children and several others in different phases of life. Many people get nervous about calculations when it’s their turn to calculate, while many are unable to solve the problem in front of others.
The scenarios can be different, but the good news is that now you can overcome math anxiety by adopting some essential tips! Do not let your child suffer from math anxiety.
Encourage a growth mindset.
Research studies have shown that effort turns ability when it comes to learning and understanding math. Encourage your child’s efforts and then set high expectations.
Get a private tutor
Tutors can have a considerable impact on a student’s feelings and mindset toward math subjects. Tutors are generally proficient at providing the personal attention that is usually hard to get at the school.
It gives the students enough opportunity to share their problems and confusions about the subject openly and boosts their confidence simultaneously.
Take baby steps
You don’t need to take the vast steps like you want to do it quickly. Take some baby steps and move slowly.
Take enough time to solve an equation, understand the problem and then practice a lot to get those results in time stringent tests.
Be involved
Students show positive and productive results when they are treated with a positive attitude and enough support.
Being a tutor or a parent, you must involve the child to do the calculations and clarify their doubts quickly to see the actual change in them.
Make maths fun
What will be worse than a child experiencing anxiety and a tedious maths teacher drowning in the mathematical equations that are complicated to understand, and you have o clue how to deal with those 50 minutes of class?
Let us consider the second scenario where studying maths is no longer a burden, and you can understand the maths concepts with fun learning activities. Interesting right!
That is what you need to do so that your child never seeks an escape whenever they get to hear the word “MATHS”. Make mathematical concepts a fun learning activity to overcome math anxiety.
Positive Reinforcement
Imagine you being bad at mathematics and your friend constantly laughs at you and your inability to solve the mathematical problems. Embarrassing, no?
But not anymore, as with positive reinforcement, you can help your child with those problems and from the upcoming embarrassment due to their inability to solve mathematical problems.
Parents and teachers both can develop positive associations with math. Instead of giving mathematical questions as a punishment for wrong answers, encourage the child to complete the work by promising a reward.
Using mixed ability grouping
It sounds embarrassing to group the class in several parts and label them with strong or low students. Confusing? You must have heard about the different grouping systems in the school of weaker students and good performers.
Indeed, a common activity is often performed in the schools, but instead of doing it this way, if the grouping is done in a manner that high and low students are grouped, it will be more helpful in boosting morale.
Yes! Grouping high with low eradicates the sense of insecurity and also lets them grow by working and learning altogether. And this is what mixed ability grouping says.
It is basically a learning method that groups together the peers having different abilities. Not every student is good in the same subject. However, mixing students well in different subjects can be a powerful way to balance the talents at equal levels and produce better results.
Turn mathematics anxiety into excitement.
If the student feels like they cannot solve a problem or get anxious while solving one, you must reframe their negative thoughts to the positive ones. Do you know how?
Well, you can do this by introducing simple real-life associations with the problem. Kids learn faster when they can relate things quickly with the daily life things that are familiar to them.
Frame some mathematical problems, associate them to daily activities, make the kids understand the association, and then give them time to solve the problems similarly. You will see a massive difference in their state of mind concerning math problems.
Encourage understanding over memorizing.
Understanding the method and memorizing by rote are the two ways considered as the learning methods.
Students generally learn by memorizing constantly to attain the lower results, while the ones understanding the concepts are likely to think outside the box while solving the math equations.
So instead of encouraging the memorizing technique, try the efficient way out of encouraging the understanding method.
Allow mistakes
Everyone who tries makes mistakes! So instead of punishing the child for their mistakes, allow them to make mistakes.
Making mistakes indicates that your child is trying and putting some effort into getting the desired outcome. Every move should be appreciated.
Be positive about math.
To understand the subject well, you must first change the perception and need to be optimistic about the subject.
Your perception influences your willingness to deal with the subject well and becomes a substantial contribution to achieving the desired outcome.
Additional Tips to overcome the math phobia
Practice math daily
Apply math in real life
Utilize resources
Study smart
Get organized
Ask questions about your doubts
Try some relaxing or breathing exercises
What can you say to yourself?
Instead of saying
Try thinking
I am not good at maths
What am I missing out on?
This is too difficult
This needs a little more time and effort
I just can’t do math
I need to train my brain in math
Oops I made a silly mistake
This mistake I am not going to repeat.
Just because it is difficult doesn’t mean you shouldn’t try; it means that you should try even harder!
Bottom Line!
Math is a comprehensive learning process! In other subjects, you learn and understand the concepts, but you understand and work on them in maths.
Every child learns differently, instead of imposing the same methods on everyone and enhancing their anxious feel concerning the subject, try to help them cope with it.
Try adopting the above-mentioned tips and support your child fighting this enemy named “MATH ANXIETY” or “FEAR OF MATHS.”
Mathematics is a nightmare for many but don’t let it be one for your child!
Lowercase ‘e’ is also known as Euler’s number named after the famous mathematician Leonhard Euler. This number is an irrational number because it has a non – recurring decimal that stretches to infinity. Euler’s number ‘e’ is a numerical constant just like pi (\pi ) and is one of the key pillars in mathematical calculations.
The value of e is 2.71828182845…………
As e is an irrational number, it is usually used under logarithm concepts. e plays a key role in problem solving in both mathematics as well as physics. The symbol e is also known as Napier’s Constant.
The value of e raised to the power 1 will give the same value as e but value of e raised to the power 0 is equal to 1 and e raised to the power gives the value as infinity. The reason is that e when results in a very big number or infinity. Similarly, when e is raised to negative power of infinity, the number turns out to be very small, which is almost equal to zero.
e is an irrational number i.e. digits after decimal are non-recurring. So, it cannot be expressed in the form of a fraction.
e has its application in many areas, concepts and subjects such as advanced mathematics and physics.
e is the base of all natural logarithms (This was discovered by John Napier).
e is transcendental or non-algebraic. This means that e cannot be obtained by performing calculations of any non-zero polynomial having finite degree as well as rational coefficients.
It’s the base to all natural logarithms i.e. Log (e) or Loge e = 1
Area under the curve y = 1/x from x=1 to x=e is 1 sq. units.
The slope of the graph of y={{e}^{x}} lies between y={{2}^{x}} and y={{3}^{x}} .
Euler’s number (e):
The Euler’s number ‘e’ is the limit of {{(1+1/n)}^{n}} as n approaches infinity. It can also be expressed as the summation of the following series.
e=\,\sum\nolimits_{{n=0}}^{\infty }{{}}1/n!
e = 1/0! + 1/1! + 1/2! + 1/3! + ………………….and so on (here, ! means factorial)
e = 2.71828182845904523536028747135266249775724709369995…. which is an irrational number upto 50 decimal places.
e is also equal to \underset{{x\to \infty }}{\mathop{{\lim }}}\,\,\left( {1+\frac{1}{n}} \right)n where n tends to infinity.
Let’s check the value of e for some numbers!
n
1
2.00000
2
2.25000
5
2.48832
10
2.59374
100
2.70481
1,000
2.71692
10,000
2.71815
100,000
2.71827
For complex numbers the formula becomes:
eiπ + 1 = 0
This number is used in various mathematical concepts and calculations. Similarly, like other mathematical constants such as beta, gamma, pi etc the value of e plays an important role.
Applications of e
Let’s check some areas where e plays a vital role not only in academics but in real life world as well.
Compound Interest
Probability, probability theorem, binomial distribution, binomial theorem and Pascal’s triangle.
Maths is often considered a feared subject for learners. Maths dread is real that makes it complicated for teachers to make mathematics learning pleasurable for learners and for students to enjoy solving it quickly.
We often hear about the stories that students and even adults try to avoid mathematical calculations. But does that mean that the people avoiding mathematical calculations are bad at mathematics? Well, not really. It’s their fear that doesn’t allow them to perform calculations confidently and which is why they avoid doing it.
The fear of maths can be overcome if a student is taught to relate mathematics with real-life examples. Things get easy to understand and memorize when people connect them with day-to-day examples.
Mathematics is often considered a hard and boring subject by many. But the experts suggest that maths can be a fun subject but only if we remove the burden from our mind that it is by default a tedious and complicated subject.
We have heard maths teachers frequently saying that doing maths questions is fun, but then we consider it an outlandish statement. Like really? How is that even possible that maths is fun? Suddenly, hearing that the most tedious and complicated subject is fun is undoubtedly a hard-to-believe statement for many! Especially when you don’t have any interest in a particular topic, you cannot concentrate for much time.
Most of all, the answer of some equations is wrong for many even after trying a lot; undoubtedly, a complicated point to understand. But things can be made more comfortable for the students by adding some smart techniques to teach mathematics.
Being a parent or a teacher, you must be worried about the child’s performance in mathematics. You must be searching for the solutions to make maths an easy-to-understand Subject. If the issue sounds familiar to you, then this blog can help to answer all your doubts concerningmathematics to be fun.
Let us take a closer look at the trending solutions to make mathematics a fun subject for your children. But before moving forward, don’t forget to analyze a few questions. Getting answers to these questions will give you a clear vision of handling things well and helping the kids better.
Are students afraid of mathematics?
Do they find mathematics a boring subject?
Do they find escape in avoiding the calculations?
Are the students conceptually weak in maths?
The answer to most of the questions is yes, and that is why the parents and teachers need to bring some innovative ideas to make it a fun activity.
7 Effective Ways Of Making Maths Fun Learning
1. Generate a Math-rich environment at school and home
Children of almost every age find mathematics a complicated subject, or the subject’s purpose might not be apparent to them. But with a mathematics-rich environment, you will be able to generate interest in students in maths.
Things get more comfortable to understand if the students can associate maths relevance with the everyday environment. Creating a maths-rich environment can be a tricky one.
You can engage students in counting books on the shelf, adding a staircase with count numbers, and letting students perform calculations by hopping on the stair after completing the given operation.
Many such ways can be an innovative way of generating interest in students. While these resources can help generate interest, it is essential to develop a number sense to familiarize the number’s vital concept. You can create the number sense in several ways. This includes:
Using sticker charts
Building blocks
Connecting shapes
Flashcards
Puzzles
Using measuring tapes, rulers, height charts
Using measuring cups, containers, and spoons
2. Play maths-related games
Every child loves to play. Instead of studying, kids love to go out and play as much as they can. But have you ever thought that you could teach them as well while playing?
If you haven’t tried it yet, then you must give it a try at least. Introduce maths games to children to generate interest in them. This can be challenging but once implemented right, the children’s maths fear won’t scare them for long.
A math-related game may include board games, card games, Uno, and many more fun games with maths calculation. The purpose of implementing this is to generate children’s interest and catch their most attention in giving accurate answers.
The world is getting smarter with mobile phones and trending applications. Several applications offer mathematics activities to the students to make their maths calculations stronger.
The way trending applications help in providing a new direction in learning is commendable. Several math applications are so quick that instead of giving the feedback and tips later after evaluations, the applications respond instantly and provide feedback on time.
The maths applications provide you with access and the ability to take your learning everywhere you go. So no more traveling with books.
4. Introduce a reward system
Children love to do the task to get a reward, and this can be a great solution to generate interest in them. But with the prize, we are not talking about agreeing to their useless demands, but you can offer the things of their interest.
You can start with the task or sum or arranging their toys by counting each within a short period or a limited time and give them a cute and encouraging reward. Something that inspires them to complete the task.
5. Concentrate on Problem-solving strategies
While talking about the maths equations, the headache of finding out the correct solution can be at its peak. Don’t teach a child always to get the right answer only; sometimes, the approach matters the most.
Focusing on strategies can help them learn better, and once their approach is correct, the chances are greater that they will tend to produce the right answer. Even if the answer is not accurate, appreciate the child’s effort to follow the strategy appropriately.
6. Visual aids
Visuals are a great way of understanding things and learning them in a better way. You can help children to understand better by representing the problem in a diagrammatic form or a chart.
Just like the Venn diagrams help the students solve the probability problems, you can add some coloring techniques to give a clear vision to the student to understand the question appropriately and figure out the solution quickly. Visuals are the saviors for understanding difficult questions easily.
7. Seek Online support
If you are done with all the math tricks to provide your child a more substantial base, and still your child is struggling with maths, then Online Tuition Supports can be a helpful solution for the problem.
Technology is a huge helping hand in providing the best education in class from the comfort of your home. The Online Maths Experts have a wide range of ways to understand maths problems and apply the appropriate solutions to them with easy-to-follow math tricks.
Consider choosing one-on-one tutoring online and give your child an unfearful maths experience.
As a parent, you must be worried about your child’s growth and academic development. Math is an essential subject that requires special attention. Try to implement the math tricks mentioned above in real life and encourage a growth mindset toward math.
If you seek an Online Tutoring Platform for your child’s growth and develop their interest in maths, then you must choose a reliable Online Platform. Settling down for a non-reliable online service provider can be a drastic step for your child’s future. This can be a life-changing step, and don’t settle for anything less.
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