How much math do we encounter daily?

How many times have we heard something similar to this phrase: "why do I have to learn about math?  I'm not going to use it anyways!"  Such a comment shocks me, because math is everywhere.

Where might we encounter math on a normal day?

1) Patterns - believe it or not, as soon as you wake up in the morning, math greets you.  If you take a look in your own bedroom, I'm pretty sure you'd be able to find some patterns, whether they are on your tiles, carpet, wallpaper, patterns are everywhere.

2) Symmetry and GEOMETRY - as soon as you're fully awake, if you're the type that loves looking out the window, you'd be able to see lots of symmetry and geometry: trees, plants, birds, even the window have symmetry and shape.

3) Time - whether you're going to work, school, or just staying home doing nothing, I'm pretty sure you take a peak at the clock several times a day.  It's just human nature.

4) Measurement - this type of math is all around us.  Your house is full of things you can measure.  Even if you're just having a meal, the amount of food intake, the numbers of calories, how much fat and cholesterol are involved, just to give few examples.

5) Money - there is no way someone can tell me that a day have passed without using money or a least seeing money.  Even if you just want to buy a cup of coffee, you have to figure out the amount of change needed for that.

6) Decimals - in a previous blog, I have described that decimals are like our shadows.  They never leave us alone.  We even have some decimals in us (i.e. weight, height, etc.).

7) Problem Solving - you might think I'm kidding, but even if you just spend your day playing video games, there is problem solving involved to figure out how to win a game.

These are only few examples of how we use and see math every single day of our lives.  I'm pretty sure now there is much more.

 

Comparing Fractions

Comparing fractions with like denominators is pretty straight forward, but once the students are introduced to fractions with different denominators, lots of confusion arises.  It's very normal for them to get perplexed, because it is sometimes not so obvious which fraction is larger than the other one.  Usually the larger the numbers, the harder it becomes to figure out the answer.  Fortunately, there is a very simple way, which helps a teacher demonstrate to his/her students how to do that comparison.

To compare any two fractions, all we have to do is cross-multiply.  That is, we multiply the nominator of the first fraction with the denominator of the second fraction, and write the value on the same side as the first fraction.  Then, we multiply the nominator of the second fraction with the denominator of the first fraction, and write that value on the same side of the second fraction.  The side with the larger value turns out to be the larger fraction.

Surely, an example would help.  For instance, to find out which one of the fractions below is larger, I will demonstrate by following the above-mentioned steps:

8/17 ? 3/7

(8*7) ? (3*17)

56 > 51

Therefore 8/17 is larger than 3/7!

After introducing such a method to the students, I'm extremely positive that the majority of the students would feel way more comfortable in dealing with fractions.

Converting a fraction to a percent

Many students find it difficult to convert fractions to percentages, because sometimes it is not so obvious what the answer is. 

The easiest and quickest way to introduce this process is to teach students how to estimate the percentage.  First, make the students get used to the idea of changing the denominator to a 100.  In order to do so, the denominator must be multiplied by a certain number.  To keep the fraction the same, we have to multiply the nominator by that same number.  Obviously, since we're working with whole numbers in this technique, the number in question is only going to be an approximation, but it helps the students get really close to the answer.  For example, if we are trying to find the percent of 8/26, we know that to get 100 as a denominator, we have to multiply both 8 and 26 by a number a bit less than 4 (since 100/4 = 25), so the percent would be slightly less than 32%.

I believe that by teaching the conversion this way, there is a good chance that students would have a better understanding about the relationship between fraction and percentage.



Caculating Percentages Mentally

Knowing how to calculate percentages mentally is not only important for kids to know, but also for adults.  How many times do we face a situation where we go to a store to buy some clothes and find out that there is a discount of a certain percentage?  If we can calculate the new value right away, we'd be able to figure out how much extra money would be left after the discount and see if we want to buy some other items with that difference.

There is a very quick way to calculate any percentage in few seconds.  All you have to do is break the percentage into two more familiar percentages (i.e. 5%, 10%, 50%, 100%).  Those percentages are more familiar, because they are very simple to calculate.  For instance, to obtain 10% of any value, we just move the decimal place once to the left, and 5% is half of that value.  To get 50% of a number, we divide by 2.

I will give couple examples to make matters clearer,

40% of 200 = ?

40% = 50% - 10%

50% of 200 = 200/2 = 100

10% of 200 = 20.0

Hence, 40% of 200 = 100 - 20 = 80

98% of 260 = ?

98% = 100% - 2% (and 2% = 10% / 5)

100% of 260 = 260

2% of 260 = 26.0/5 = 5.2

So, 98% of 260 = 260 - 5.2 = 254.8

This technique demonstrates that percentages are very easy to calculate, so long as we keep things simple.

Squaring numbers is as easy as 1, 2, 3!

In my previous blog, I have talked about a trick to square numbers falling in the range 30-70.  I mentioned, however, that because of its limitations, many people would rather try to find a more clever trick that would involve a wider range.  I'm going to illustrate here another trick, which is way more efficient, because it could be used for all numbers.

In order to square any number in a matter of seconds, all you have to do is follow these simple steps:

1) calculate the difference between the number you are trying to square and the nearest base (10, 100,  1000, etc.)

2) Add that difference to the number to be squared to get the first digits

3) Square the difference to get the rest of the digits

4) Don't forget to carry over!

For example, if you are trying to square a big number, such as 123, what you have to do is:

1) the nearest base is 100, so 123-100 = 23

2) 123 + 23 = 146

3) 23 x 23 = 529

4) 14(6+5)29 = 14(11)29

carry over again: 1(4+1)129 = 15129 - TADA!!

Wasn't that simple?  This technique works with all numbers if you follow the above steps carefully, so who needs calculators?!

Squaring numbers between 30 and 70

Most of us panic if asked to square a number larger than 12 and get out a calculator right away.  It turns out that this does not have to be the case for all situations, because there are some very bright individuals out there that are coming up with "math magic" for us to use.  Those individuals fully realize how lazy the human brain could be, and that if a process takes so long, then we would not approve of it anymore.  Unfortunately, this trick is limited, but the good news is that it works perfectly and is quick and easy.

The first thing to keep in mind is that this process only works for numbers between 30 and 70.  Why you might ask.  Simply because it relies on the number 50.  So anytime you have a number in the above-mentioned range, here is what you have to do:

1) To get the first two digits, work out the difference between that number and 50, and

     a) if it's less than 50, subtract the difference from 25

     b) if it's more than 50, add the difference to 25

2) To obtain the other two digits, square that difference and remember that if it's less than 10, for example if it's 4, you write it as 04. 

3) If the square of the difference is a three digit number, the idea of carrying over applies.

So how does all that look like in an example?  For instance, to get the value of 62 squared,

1) 62 - 50 = 12 and 25 + 12 = 37

2) 12 x 12 = 144

3) since 144 is a three digit number, we have to carry the 1 over to 37: (37+1)(44)

Hence, the answer is 3844.

This trick is an amazing one, but because of its limitations, I'd rather stick with another one that applies to all numbers.

Multiplying by two digits mentally

As a child, I was amazed at the speed some people performed some long mathematical processes.  I did not know back then that there are so many nice tricks in math.  One of my favourites is the way an individual can obtain the answer of multiplying any big number by two digits as fast as five or six seconds.  So how does it work?

The simplest one to start with is multiplying any number by 11.  I'll start with an example to simplify matters.  So what if we were asked to get the answer to the following: 45326 x 11?  All we have to do is:

1) copy the first number as it appears

2) add each two digits starting with the first one at the far left to get the numbers in the middle

3) copy the last number as it appears

Hence, the answer would be: 4(4+5)(5+3)(3+2)(2+6)6 = 498586.

You might ask, though, what would happen if the sum is larger than 10?  Simply use the idea of "carrying over".  For example, what is 318432 x 11?

318432 x 11 = 3(3+1)(1+8)(8+4)(4+3)(3+2)2

= 3(4)(9)(12)(7)(5)2

We now how to carry over the "1" from the 12 to the 9.  So the answer becomes:

34(9+1)2752 = 3502752

What happens then if we're multiplying by a two-digit number different than 11?  Same story, but all we have to keep in mind is that there is an extra step involved.  Again, it would be best to illustrate with an example:

24321 x 13 = 2[(2*3)+4][(4*3)+3][(3*3)+2][(2*3)+1](1*3)

= 2(10)(15)(11)(7)(3)

= 316173

So the only difference here is that we multiply the first number in each sum and the very last digit by the "3", which is the "1's" digit in 13 and then proceed as before!!

This trick proves that math could be not only fun, but very easy to understand!

Is Long Division Important?

Long division is a very old concept that is almost forgotten, because many students now rely on calculators.  Some people might argue that it is irrelevant for children to learn about it, and that it could be very confusing.  I feel that all kids should be introduced to it, because it helps them understand the definition of division.  Long division is only confusing when being taught the wrong way. 

In order to help kids understand this process, it is of utmost importance to use base 10 manipulatives.  It should not be a big deal if kids do more steps at the beginning so long as they understand the concept.  The best way to illustrate how to do long division is dividing the given number into 100's, 10's and 1's.  For example, if we're trying to figure out how many times 3 goes into 36, the students should first build the number 36 as three 10's and six 1's.  Next, they have to divide those 10's and 1's into three equal groups.  It turns out that each group would have one 10's and 2 1's, so the answer would be 12.  Once the students grasp the idea by using manipulatives, they can start showing their work on paper.  The teacher must encourage them to still use notations showing 100's, 10's, and 1's.  This process might sound tedious and long at first, but once students are comfortable with it, things become much better.

We should never discourage a kid to learn about long division.  It is a straight forward process if taught correctly and students can benefit a lot after being introduced to it.



Beauty of Decimals

There is no reason why anybody should be confused about decimals because they are everywhere in this world.  Whether we measure our height, weigh ourselves accurately, deal with prices, make a trip and figure out how many kilometers we've traveled, decimals are our companions day and night.  They live with us and share our good and bad moments.  I view decimals as shadows, because no matter how hard we try to get away from them, they follow us.  Yet, many confusion arises when introducing this concept.

In order to be successful in explaining decimals to students, a teacher should first make them aware how much we use decimals every day by making up a simple scenario.  For example, the teacher could say:

"I would like to share something that happened with me last night.  After I left the school and traveled for 10.7 kms, I realized that I have promised my best friend to go to the movies.  Since I was going the wrong way, I had to turn around and travel 5.3 kms the opposite way.  I met my friend and we bought the movie tickets, and some popcorn and coke.  That costed us $32.87 altogether.  After the movies, we decided to have dinner together.  The menu at the restaurant we went to showed dishes ranging from $10.60 to $43.25, etc.."

Listening to such a story helps students realize how decimals are everywhere, and hence they would want to know more about that topic.  Once this is all done, the teacher could play a "restaurant game" with the students, by giving the students a menu where they have to figure out what food they can order based on the amount of money they have.  That way, they become familiar with adding decimals and figuring out which ones are larger than others.

I believe that decimals should be one of the very first concepts to be introduced to students.  As soon as we help kids realize how much they use them already, it would easy for them to grasp the idea and become decimal experts.

Keeping Students Interested

Every human being is willing to pay attention to a topic that is related to his/her interest.  For instance, individuals who love sports would usually read that section of the newspaper first, and then concentrate on the other topics, which wouldn't be equally interesting in their opinion.  So it becomes highly important for teachers and parents to figure out every child's interest, and take advantage of that when trying to teach them a new concept.

On of the crucial steps that a teacher has to go through for a successful year ahead is spend some time learning every student's interest at the beginning of an academic year.  The teacher then, would be able to introduce lessons and concepts based on what they enjoy best.  For example, once a teacher finds out that many students in his/her class enjoy playing video games, he/she can encourage them to play certain games that enable them to learn whichever concept he/she is covering at school.  There are many games, for instance, that teaches kids how to skip count, add, subtract, etc.  If there are various interests in the classroom, the teacher could give various examples, each of which concentrating on one of them during the lesson, or even in practice exercises.

We should always keep in mind that kids only excel in something they really love.  With a bit of creativity, we would be able to make learning much more enjoyable.

Connecting Math Concepts to Real Life

As a novice in the teaching field, I always try many different ways of delivering a lesson to find out which ones are more successful than others.  It turned out that if lessons do not include real life situations in them, students eventually lose interest and start daydreaming.

It is not about children being stubborn or picking and choosing what they want to learn about.  Kids are smart, and they refuse to exercise their brain muscles for concepts that they view as irrelevant and would not serve them any good in their practical lives.  Hence, teachers and parents play an important role in the process of introducing real life math to children.  They should take advantage of all opportunities to point out how math is used in their everyday's life. 

For instance, if a kid is trying to save some money to buy a toy by doing chores, the parent(s) should help him/her figure out how many chores are required (addition), or at a later stage, the amount left to be able to buy that toy (subtraction). 

Road trips could be great opportunities for children to relate speed and travel time.  If the child is at a young age, the parent can ask him/her to keep track of how many stop signs they have encountered, or how many different car colours they have passed by, etc. 

Sharing between siblings can help them learn about divisions and fractions by introducing the idea of fairness.  By splitting a bag of chips, for example, among themselves into equal parts, division becomes a natural process.

Even when a parent is grocery shopping, they can ask their kids' help in figuring out how much money their list is going to cost.  They can even calculate if the money would be enough based on a specific budget (addition, multiplication, subtraction, and money concepts).

These are just few ideas that facilitates the basic math concepts for children.  There are many other situations that could help kids learn math without even realizing it.



Active Math

Being a mother of three, I discovered that children love to move around all day long and find it very boring if they have to sit down for a long time to do an activity.  No matter how interesting that activity is to them, if it requires lots of sitting, they would rather give it up and do something more active.  Knowing that fact, we should be careful in how we teach those youngsters.  Physical activities must be included in all our lessons.

Some might find it impossible to teach math while keeping kids active.  We all think of math as a worksheet given to students to practice while sitting down.  Many educators are trying to change this belief now by icluding movements to teach many basic math concepts.  For example, the founder of a website called "math and movement", Suzy Koontz, illustrates many ideas where math and movement are done together.  That program proved that children, as young as pre-k, can participate in activities that enables them to learn about addition, subtraction, telling time, skip counting, and multiplication.  Such programs are successful because research have shown that moving while learning facilitates muscle memory.  It was also proven that cross body movement (i.e. when the left arms or legs are crossed over to the right side of the body or vice versa) integrates the left and right areas of the brain.

Hence, it becomes very important for every teacher or parent to engage kids in daily physical activities.  So why not use the children's natural abilities to enhance their love of learning!

Cooks are great mathematicians

Some people insist that they are horrible in math no matter how hard they try.  The truth, however, is that many people, especially those that enjoy cooking, are great in math because they use it everyday without even realizing it.



1) Most cooks like to buy the ingredients themselves, and that is the first step of using math.  When selecting the items to be used, they have to figure out how much of each to get according to the recipe they're using. 

2) If they're working on a budget, they have to figure out if the amount of money they would like to spend would be enough to buy all necessary items.

3) Once all the above is taking care of, they then have to modify the ingredients according to how many people they're going to serve, and that usually involves FRACTIONS!!

4) To use the utensils in the kitchen, lots of conversions are involved.  For example, how big is a cup?  How many teaspoons are in a tablespoon?  How many tablespoon in a 1/4 cup?  How many cups are in a gallon?  etc.

5) If they're tight on time or would like to serve the food as soon as it's done, they have to figure out how long the cooking process is going to take and inform their guests.

6) They have to figure out what temperature to set the stove on, if they're preparing something that requires using the stove, and they have to know how to convert between celsius and fehrenheit.

So next time you meet a cook, appreciate how much math they know!

Fraction Tools

Many students are scared of fractions because of how confusing they could be.  Some of them even have a hard time identifying that a 1/4 is smaller than a 1/3.  Fortunately, there are many tools that teachers could use to simplify this concept.  The most important thing, however, is not to tell the students that they will be learning about fractions at first.

Some of the most popular tools that a teacher could use when teaching fractions are:

1) Pizza - of course, one of the yummiest ways of learning about fractions is to bake a pizza for the students and cut it into pieces to illustrate the basic ideas of this concept.

2) Virtual manipulatives - if a computer is accessible at the school, there are many virtual manipulatives that could be used, and most of them are vey powerful to make matters easy and enjoyable.

3) Fraction disks - students are able to do so many activities with these colourful disks.  They show fractions as wholes, halves, thirds, fourths, fifths, sixths, sevenths, eighths, nineths, and tenths.  These are perfect for group activities.

4) Fraction wheels - these wheels usually go up to 1/10 and are perfect for students working on individual activities.

5) Fraction sets - sets of different fraction of rectangle and square are put together for the students to feel what different fractions look like and why equivalent fractions are the same.

6) Fraction bars - these manipulatives are solid tiles that represent a whole, halves, thirds, fourths, fifths, sixths, eights, tenths, and twelfths.  These are perfect for teaching the students the difference between nominators and denominators.

In conclusion, fractions could be scary and confusing only if we allow them to be so.  With all the different tools available, teaching and learning fractions could be lots of fun for teachers and students, respectively.

Dividing by Zero

We have all learned as students that dividing any number by zero is undefined, but why is that so?  To understand this, we first have to define division. 

So what is division really?  Division is splitting a number into equal groups.  For example, if three children are fighting over a bag of candy that contains nine pieces, we would split that bag into three groups of three, so that we're fair to all three kids.  Knowing that then, to divide by zero would mean to split a number into zero groups.  To go back to the idea of children fighting over candy, would we be able to split that bag into zero kids?  Does that even make sense?  Not quite! 

The other nice thing about division is that we can always check our answer by multiplying the two numbers to get the third.  For example, if we just figured out that 10 divided by 5 is 2, we can multiply 5 and 2 to verify that the answer is indeed 10.  Having zero, that concept does not work:

10 divided by 0 DOES NOT EQUAL 0 divided by 10

Hence, dividing by zero is undefined simply because it doesn't have any mathematical or logical sense.

Simplest Way to Teach Multiplication

Multiplication can be a hard concept for some students to understand if they are asked to memorize a multiplication table, simply because they would not be able to grasp the main idea behind it.  If we teach the students that multiplication is a repeated addition, things become much clearer. 

While manipulatives can easily be used, I find that cookies are a better fit for that.  Most, if not all, students love chocolate chip cookies, so why not use something they're used to seeing and eating as a tool of learning.  For instance, the teacher can show the students that to get the value of 2 x 5, it means that we have to have two groups of fives (or even five groups of twos).  He/she can draw two cookies and show five chocolate chips on it.  The students would be encourage to count the chocolate chips and find the total, which would be 10.  Another example could be shown to make sure everyone understands.  To obtain the value of 4 x 3, for example, four cookies should be drawn with three chocolate chips on each, for a total of 3+3+3+3 = 12.

I believe that there is no better way of illustrating multiplication other than using cookies, so go ahead start baking those chocolate chip cookies!

Base 10 Manipulatives



I was never introduced to base 10 manipulatives as a child, even though they have existed for a long period of time.  Maybe it depends on what country we grow in, or maybe it's just a teacher's preference.  I'm really not sure, but even now I meet some teachers that aren't very comfortable to use them in their classrooms.  I don't know why many mentors prefer not to use them, even though they can be very powerful in explaining many things in Math. 

When I recently got introduced to the base 10 manipulatives in one of our University Education classes, and by the way I did not know what they were called, I was shocked and I wasn't sure how using "toys" would be beneficial in Math.  Our instructor's enthusiasm and ease of guidance helped me realize that it would be a waste not to use them for various reasons.

1) It is easier for students to remember things and learn about them if they've seen them and have dealt with them rather than just heard about them.  For example, if students are learning about subtraction using base 10 manipulatives, the process becomes very clear in their mind when they actually see what happens step by step.

2) Base 10 manipulatives help students stretch their imagination.  With that aid, they would be able to relate what they're learning to real life situations. 

3) Problem solving becomes a second nature for students who use manipulatives, because in most cases, it is likely that there is more than one solution available.  They will be able to successfully choose the best strategy in each situation.

I was very happy and thankful to be introduced to the base 10 manipulatives.  If I ever have my own classroom, I will definetly use them wherever is applicable.

Best Way of Solving the 12 Days of Christmas Problem

I have mentioned in my last journal that one of the approaches of calculating the total amount of presents in the 12 days of Christmas song is to use Pascal's triangle, but that could be tedious for more challenging situations.  Knowing that the binomial coefficients are "specific addresses" in Pascal's triangle as shown in the figure, it becomes way easier to solve problems similar but more complicated than the 12 days of Christmas. 

Here is how it works:

The binomial coefficients are written in any form similar to "n choose k".  That refers to the kth element in the nth row in Pascal's triangle.  To verify, one can use the formula first and then compare with the corresponding value in Pascal's trianlge.

So if we, for example, are trying to choose two items out of seven options, the answer according to the formula is:

and sure enough, if we look at the 2nd element in the 7th row in Pascal's triangle (keeping in mind that the first row in Pascal's triangle is the 0th row), the value is 21.

Hence, the nice relationship between the binomial coefficients and Pascal's triangle helps a lot in solving more complex problems.

The 12 Days of Christmas Song & Pascal's Triangle

The 12 days of Christmas song is a popular exercise given to students in grades 5 or 6 to calculate how many presents in total are given during a 12-day period.
This question can be solved in many ways.  One very nice approach is to solve it using Pascal's triangle as follows: 
The first diagonal to the left of the 1's (i.e. 1,2, 3, ...) gives the number of new gifts given on consecutive days (i.e. one partridge in a pear tree, two turtle doves, etc.). 
The second diagonal to the left of the 1's (i.e. 1, 3, 6, 10, ...) gives the sum of the presents on consecutive days (i.e. 1 = 1 partridge in a pear tree, 3 = 2 turtle doves + 1 partridge in a pear tree, ...).
The third diagonal to the left of the 1's (i.e. 1, 4, 10, 20, ...) gives the sum of the presents given.  For example, on day 3, 10 presents are given in total (3 French hens +(2*2 turtle doves) + (3*1 partridge in a pear tree)), and that indeed matches the third number of that diagonal.  So on day 12, the 12th number of that diagonal, which is 364, represents the total number of presents received.
In conclusion, Pascal's triangle solves the problem of finding the sum of those presents, but it could become very tedious for more challenging situations.  For instance, what would happen if the song was about giving presents for 120 days instead of 12?  In that case, we would have to come up with a better approach for sure!

Dealing with "Silly" Mistakes

How many times have we told a student something similar to the following: "Come on, stop doing those silly mistakes and concentrate harder!!"?  Do such harsh words usually help the student focus more?  Most likely not!  All we're actually doing is lowering their self esteem.

I believe that no one intentionally errs.  There are reasons for everything we do.  So what is really behind those "silly" mistakes?  I would like to discuss few of them here:
 
a) One of the basic sources of such mistakes is what I like to call "foundation problem".  If there is a basic theory that the student does not understand, that would create many problems.  So it's always important to diagnose the mistake carefully and try to figure out what is causing it.

b) Another reason would be if the student is working too fast.  Solving problems hastily could be a dilemma for many students, because they are not taking their times to think carefully about a problem and double check their work.  It's of utmost importance to ask a student to take his/her time while solving a problem.

c) Sometimes the type of food we consume affects the way our brains function.  It's important to pay close attention to find out if there is a constant occurence of confusion because of a specific food being eaten.

d) The intake of water could be another cause.  Many research have proven that lack of water causes the short memory of the brain to become weaker.  We always have to ensure that we're drinking lots of water to avoid that problem.

These are just few of the reasons accountable for "silly" mistakes.  So before accusing someone of being stupid or silly and hurting his/her feelings, let's focus on what is creating that confusion.