As I was walking this morning in my neighborhood, my mind was filled with thoughts that geometric shapes are everywhere! Do you agree? Gazing at sidewalks, homes, street signs, and security lights, I saw shapes of rectangles, trapezoids, triangles, octagons, and even circles and spheres! Try it sometime, you'll be amazed how many hundreds of shapes you see each and every day!
For today, let's take a look at the clock. Other than public buildings, such as schools and medical offices, do you have a clock in your home? Our society has quickly replaced wristwatches with digital forms of knowing the time on cell phones, tablets, computers, and other electronic devices. Most of them are not displayed in the shape of a circle, but are digital. Perhaps some of you reading this right now never learned the time of day by reading a clock, but have been exposed to only digital displays of time.
There are some easy lessons to learn from the clock. What traditional shape is a clock? It is a circle, isn't it? If you are not familiar, there are numbers from 1 through 12 on the clock, and we use it to measure all 24 hours in the day. The hands on the clock spin around quickly, quietly, and smoothly. From the 12 to the 1, the hour hand indicates that 1 hour has just passed. From the 12 to the 3, the hour hand has moved 3 hours. From the 12 to the 9, the hour hand has moved 9 hours. Each day, there are 2 times that it is 1 o'clock, 1:00am and 1:00pm. There are also 2 times each and every day it is 6 o'clock, and 7 o'clock, and 8 o'clock,...etc.
The natural movement of the hands on a clock to the right is called clockwise. If the hands were to move in the opposite direction, it is called counter-clockwise. When you assemble furniture, or repair your car, it is common for the instructions to guide you to turn the tool clockwise or perhaps counter-clockwise.
For the purpose of our lesson, we have assumed that our clock is a circle. How many degrees are in a circle? That's right! There are 360 degrees in a circle. Since there are 360 degrees in the entire circle, and there are 12 hours in the circle, how many degrees are in each hour? We divide 360 degrees by 12, and the answer is 30 degrees. We know that for each hour that passes, there are 30 degrees that have passed also, each day, every day, all day long.
Since we know there are 30 degrees in each hour, we can also determine how many degrees there are in either direction from one hour to another on the clock. Yes, we can do it both clockwise and counter-clockwise!
Let's try a few examples:
1.) How many degrees does the clock travel clockwise from 1 o'clock to 4 o'clock?
Solution:
* There are 3 hours from 1 o'clock to 4 o'clock.
* We know there are 30 degrees each hour.
* Simply calculate the product of 3 x 30 = 90 degrees.
That is correct! There are 90 degrees from 1 o'clock to 4 o'clock.
2.) How many degrees are there from 5 o'clock to 1 o'clock going counter-clockwise?
Solution:
* There are 4 hours from 5 o'clock to 1 o'clock.
* Remember there are 30 degrees per hour.
* Calculate the product of 4 x 30 = 120 degrees.
You are correct, once again! There are 120 degrees from 5 o'clock to 1 o'clock.
3.) Let's do #2 again, but this time we need to calculate the number of degrees from 5 o'clock to 1 o'clock traveling clockwise.
Solution:
* There are 8 hours from 5 o'clock to 1 o'clock.
* Use the 30 degrees per hour measurement once again.
* Multiply 8 x 30 = 240 degrees.
That's right! You are quickly mastering your knowledge of the degrees in clock!
Try a few of these on your own! They are fun and very practical! Don't forget to share this great information with a family member or friend!
Sunday, May 21, 2017
Friday, May 19, 2017
Mental Multiplying--You Can Do This!
Are you like most people when you hear the words 'Mental Math' you immediately rebel and ask yourself, "Why mess with that?" Or perhaps you might be thinking it is far too difficult and there is no advantage for me, right?
You could be at the market, and want to know very quickly how many units of something you can buy with the cash you have. Maybe you are an attorney, and you need to do some quick Math while defending your client. Whatever the situation, the quicker you can think on your toes, the greater the advantage you definitely have.
Let's take a look at the basics of multiplying a couple numbers in your head, and without using the calculator on your cell phone, or paper and pencil.
Evaluate 4(96)
Using the Distributive Property, think of a couple different ways to add or subtract two numbers to yield 96.
Solution 1:
4(100 - 4)
4(100) - 4(4)
400 - 16
384
Solution 2:
4(90 + 6)
4(90) + 4(6)
360 + 24
384
Which solution did you prefer? Did you notice that both solutions gained the same answer? It is important to proceed with your solution using numbers that are easy to multiply in your head, so we are referring to numbers that are being multiplied by a multiple of 10, and possibly multiples of 100 or 1000, in addition to performing simple calculations from numbers that are a part of the basic multiplication tables that most of us memorized in elementary or middle school.
Try it, and with a little practice, you will be an expert very quickly! Don't forget to teach this great tip to a family member or friend! If you take a few minutes to explain it to someone, you will increase your own skills and gain much confidence!
Happy Multiplying!
You could be at the market, and want to know very quickly how many units of something you can buy with the cash you have. Maybe you are an attorney, and you need to do some quick Math while defending your client. Whatever the situation, the quicker you can think on your toes, the greater the advantage you definitely have.
Let's take a look at the basics of multiplying a couple numbers in your head, and without using the calculator on your cell phone, or paper and pencil.
Evaluate 4(96)
Using the Distributive Property, think of a couple different ways to add or subtract two numbers to yield 96.
Solution 1:
4(100 - 4)
4(100) - 4(4)
400 - 16
384
Solution 2:
4(90 + 6)
4(90) + 4(6)
360 + 24
384
Which solution did you prefer? Did you notice that both solutions gained the same answer? It is important to proceed with your solution using numbers that are easy to multiply in your head, so we are referring to numbers that are being multiplied by a multiple of 10, and possibly multiples of 100 or 1000, in addition to performing simple calculations from numbers that are a part of the basic multiplication tables that most of us memorized in elementary or middle school.
Try it, and with a little practice, you will be an expert very quickly! Don't forget to teach this great tip to a family member or friend! If you take a few minutes to explain it to someone, you will increase your own skills and gain much confidence!
Happy Multiplying!
Thursday, June 16, 2016
Friday Funny! 😂😂😂
Have you ever had a Math book fall on your head? Who else is there to blame but your shelf?
😂😂😂😂😂😂😂
Happy Friday, everyone!
😂😂😂😂😂😂😂
Happy Friday, everyone!
Saturday, June 11, 2016
What Is My Approximate Annual Salary?
Without performing the messy calculations, is there a quick way to know approximately how much you will earn in a year if you work the entire year and average 40 hours per week? Many years ago, I learned a trick, and the process remains a valid one today.
Let's take a look at a possible real-world scenario, and understand the benefit of knowing this helpful information.
Mr. Miller works 40 hours each week throughout the year and currently makes an annual salary of $40,000 and would like to find a job that pays more money without working more hours. He has recently interviewed for a job at the local factory that pays $25 per hour and he is trying to determine which job will earn more money?
Here is the solution:
Traditional Method of calculation:
1.) Multiply $25/hr x 40 hrs/wk x 52 wks/yr.
2.) He will earn $52,000.
Alternate Method of Approximation:
1.) Multiply the hourly wage $25 and double it, which is $50.
2.) Multiply by $1,000 or if you prefer you may add 3 zeros, which is $50,000.
Let's take a closer look at the math operations and gain a better understanding of how this actually works. If he works 40 hrs per week every week and there are approximately 50 weeks in a year, we can quickly multiply 40 x 50 and know the product is 2,000. We also know that 2 factors of 2,000 are 2 x 1,000 which enables us to double the hourly wage (utilizing the 2 of both factors) and apply the 3 zeros in 1,000 to the product, which gives us the estimated wage of $50,000.
Some of you may be concerned that the wage varies by $2,000 from the actual calculated amount, and my reasoning for accepting this method of approximation, is that you know you'll make at least $50,000 and it would be wise for Mr. Miller to accept the new job offer if he is ready to make a change in his employment.
One quick note about the variance in the calculations is knowing that the higher the hourly wage, the greater the variance will be between both annual salaries. Another note about the difference in wages, by underestimating the actual wage, we are offering a layer of financial protection by accounting for 50 of the 52 weeks in the year.
Try a few examples on your own, and you will soon be the expert! 😉😉
Let's take a look at a possible real-world scenario, and understand the benefit of knowing this helpful information.
Mr. Miller works 40 hours each week throughout the year and currently makes an annual salary of $40,000 and would like to find a job that pays more money without working more hours. He has recently interviewed for a job at the local factory that pays $25 per hour and he is trying to determine which job will earn more money?
Here is the solution:
Traditional Method of calculation:
1.) Multiply $25/hr x 40 hrs/wk x 52 wks/yr.
2.) He will earn $52,000.
Alternate Method of Approximation:
1.) Multiply the hourly wage $25 and double it, which is $50.
2.) Multiply by $1,000 or if you prefer you may add 3 zeros, which is $50,000.
Let's take a closer look at the math operations and gain a better understanding of how this actually works. If he works 40 hrs per week every week and there are approximately 50 weeks in a year, we can quickly multiply 40 x 50 and know the product is 2,000. We also know that 2 factors of 2,000 are 2 x 1,000 which enables us to double the hourly wage (utilizing the 2 of both factors) and apply the 3 zeros in 1,000 to the product, which gives us the estimated wage of $50,000.
Some of you may be concerned that the wage varies by $2,000 from the actual calculated amount, and my reasoning for accepting this method of approximation, is that you know you'll make at least $50,000 and it would be wise for Mr. Miller to accept the new job offer if he is ready to make a change in his employment.
One quick note about the variance in the calculations is knowing that the higher the hourly wage, the greater the variance will be between both annual salaries. Another note about the difference in wages, by underestimating the actual wage, we are offering a layer of financial protection by accounting for 50 of the 52 weeks in the year.
Try a few examples on your own, and you will soon be the expert! 😉😉
Monday, June 6, 2016
Tropical Depression or Tropical Storm?
Since I was a young boy, I have always been fascinated with weather, and particularly watching storms brew. Living in Indiana most of my life, I've never experienced a Tropical Depression or Tropical Storm, but we've certainly had numerous storms with winds that were in excess of 60 mph, that often produced heavy downpours and often hail.
According to the many weather experts, a Tropical Depression is when a storm delivers sustained winds of 38 mph or less, and if the sustained winds are 39 mph or greater, it becomes a named storm and is changed to a Tropical Storm.
As of today, Tropical Storm Colin will be making a path from the southwest waters in the Gulf of Mexico and will pass over Tampa in a northeasterly direction across Florida and then aim for the east coast.
Be safe everyone, and take this storm very seriously, as extensive flooding may occur. This storm may produce tornadoes as well.
According to the many weather experts, a Tropical Depression is when a storm delivers sustained winds of 38 mph or less, and if the sustained winds are 39 mph or greater, it becomes a named storm and is changed to a Tropical Storm.
As of today, Tropical Storm Colin will be making a path from the southwest waters in the Gulf of Mexico and will pass over Tampa in a northeasterly direction across Florida and then aim for the east coast.
Be safe everyone, and take this storm very seriously, as extensive flooding may occur. This storm may produce tornadoes as well.
Saturday, June 4, 2016
How Fast Do Greyhounds Run?
Many years ago, I attended a greyhound race, and wondered how fast these fascinating dogs run. If I were to tell you that some of the faster greyhounds can sprint as fast as 60 feet per second, do you know approximately how many miles per hour they can run?
Let's take a look at an example.
How do you convert feet per second to miles per hour?
Solution:
60 feet 60 seconds 60 minutes 1 mile 216,000
--------- x ------------- x ------------- x ----------- = ---------- = approx. 40 miles/hr
second minute hour 5280 feet 5280
To solve this problem, you need to multiply 60 x 60 x 60, which is 216,000. One more calculation, divide 216,000 by 5280, and arrive at the answer, approximately 40 mph.
There you have it, a greyhound runs as fast as 40mph!
Let's take a look at an example.
How do you convert feet per second to miles per hour?
Solution:
60 feet 60 seconds 60 minutes 1 mile 216,000
--------- x ------------- x ------------- x ----------- = ---------- = approx. 40 miles/hr
second minute hour 5280 feet 5280
To solve this problem, you need to multiply 60 x 60 x 60, which is 216,000. One more calculation, divide 216,000 by 5280, and arrive at the answer, approximately 40 mph.
There you have it, a greyhound runs as fast as 40mph!
Tuesday, May 31, 2016
How Much Water to Fill My Pool?
Do you ever wonder how many gallons of water it would take to fill a medium size above- or below-ground swimming pool? If you are similar to the average person, you have likely thought about it once or twice in your lifetime, but probably not much more than that. Perhaps during one of your famous cannonball dives and several gallons are displaced out of the pool, you may have thought about it, in a competitive way!
The formula is fairly simple, especially if you are familiar with a few Math terms. Let's look at a couple examples.
1.) How many gallons of water does it take to fill a circular swimming pool that measures 20ft across and has an average water depth of 5ft?
Solution:
* Find the area of the surface, which = Pi (3.14) x Radius squared.
3.14 x 24 x 24 = 576 sq.ft.
* Find the volume of the pool by multiplying the top surface area times the depth of the pool, which is 5 ft.
576 sq.ft x 5ft = 2880 cubic ft.
* Knowing that there are approximately 7.5 gallons of water per cubic foot, you can multiply the volume times the number of gallons of cubic feet to know how many gallons you need.2880 cubic ft. x 7.5 gallons per cubic ft. = 21,600 gallons!
Take a look at another example, this time the pool is rectangular shape, and measures 16ft x 36ft and is 5ft deep throughout the pool. How much water does it take to fill the pool?
Solution:
* Find the area of the surface which is 16 x 36 = 576
* Find the volume of the pool by multiplying the top surface area x the depth, which is 576 x 5 = 2880 cubic ft.
* Multiply the volume x 7.5, which is 2880 x 7.5 = 21.600 gallons!
Did you notice something? Both pools with different shapes and different dimensions require the same number of gallons to be filled! Next time you are trying to decide which pool to build at your new home, you will know that there is more than one shape of a pool that can have the same overall dimensions and water usage!
For those of you who want to know how many Liters it would take to fill the pools in our examples, you can simply multiply 21,600 gallons x 3.78 L per gallon. It would take 81,648 L to fill each pool!
Happy swimming, everyone!
The formula is fairly simple, especially if you are familiar with a few Math terms. Let's look at a couple examples.
1.) How many gallons of water does it take to fill a circular swimming pool that measures 20ft across and has an average water depth of 5ft?
Solution:
* Find the area of the surface, which = Pi (3.14) x Radius squared.
3.14 x 24 x 24 = 576 sq.ft.
* Find the volume of the pool by multiplying the top surface area times the depth of the pool, which is 5 ft.
576 sq.ft x 5ft = 2880 cubic ft.
* Knowing that there are approximately 7.5 gallons of water per cubic foot, you can multiply the volume times the number of gallons of cubic feet to know how many gallons you need.2880 cubic ft. x 7.5 gallons per cubic ft. = 21,600 gallons!
Take a look at another example, this time the pool is rectangular shape, and measures 16ft x 36ft and is 5ft deep throughout the pool. How much water does it take to fill the pool?
Solution:
* Find the area of the surface which is 16 x 36 = 576
* Find the volume of the pool by multiplying the top surface area x the depth, which is 576 x 5 = 2880 cubic ft.
* Multiply the volume x 7.5, which is 2880 x 7.5 = 21.600 gallons!
Did you notice something? Both pools with different shapes and different dimensions require the same number of gallons to be filled! Next time you are trying to decide which pool to build at your new home, you will know that there is more than one shape of a pool that can have the same overall dimensions and water usage!
For those of you who want to know how many Liters it would take to fill the pools in our examples, you can simply multiply 21,600 gallons x 3.78 L per gallon. It would take 81,648 L to fill each pool!
Happy swimming, everyone!
Monday, May 30, 2016
No Boss Will Ever Say This!
I saw something today, and it caused me to realize once again why I do what I do, but even more importantly, that I truly enjoy being a Math Teacher and helping others learn.
The sign read, "I need to hire employees who can't do Math--said no boss ever!"
Family and friends, it is never too late to learn new skills, or to brush up on the ones that you need to re-learn. I have created a new blog today helpmedothemath.blogspot.com which is a blog similar to this one, but you can directly ask me a Math question and receive help.
The sign read, "I need to hire employees who can't do Math--said no boss ever!"
Family and friends, it is never too late to learn new skills, or to brush up on the ones that you need to re-learn. I have created a new blog today helpmedothemath.blogspot.com which is a blog similar to this one, but you can directly ask me a Math question and receive help.
Saturday, May 28, 2016
Receive Change Like a Boss!
Do you go to the market or restaurant, and frequently receive a large number of coins and small bills, but wish you knew the trick to receive larger denominations of both? I remember as a youth watching my mom and dad make purchases at the market and be able to adjust the amount they received in change, quickly knowing how to make an adjustment to the amount they gave the cashier to pay for their groceries.
In my adult life, I have been instructed by a cashier from time to time that it was too late to make an adjustment to the amount that he/she had expected to receive from me, primarily in the form of a whole dollar bill, perhaps a $5, $10, or $20. I have been told by cashiers also, that they have been advised to provide the exact amount of change that the 'computer told them to give' and to not deviate from it. The trick and very first step is being confident and ready and to not show the cashier the amount you are paying until he/she is ready to receive your payment.
Let's take a look at a real-world example and see how you can adjust the amount of change you receive.
You are at Dave's Deli and have purchased a turkey sandwich and a large drink. The total due is $7.36, and you are paying cash for your purchase. Let's assume you have a $20 bill, a $10 bill, and (3) $1 bills, and the following coins: (3) Quarters, (1) Dime, and (4) Pennies. There is more than one solution to the amount that you may give Sondra, the deli's ace cashier.
Knowing that you don't have the exact change to give, here are some possible scenarios:
a.) Simply give her a $10 bill, forgetting that you have several coins in your pocket, and expect to receive $2.64 in change from Sondra.
b.) You remember that in addition to the $20, $10 and few $1 bills, you have several coins in your pocket. You quickly decide to pay her $10.36 and she returns $3 in one dollar bills.
c.) Sondra sees that you have some pennies in your hand, and she asks you to give her one penny in addition to the $10, making the total amount received $10.01. You are thinking to yourself, "Why is she asking me for an extra penny, when $10 is more than enough to cover your sandwich and drink?"
Sondra, seeing the confused look on your face, quickly explains that instead of returning $2 and $0.64 in coins, by giving her the extra penny, she now owes you change of $2.65 which simplifies the Math, saves her 4 pennies for the next customer, and you don't have to carry around extra coins in your pocket.
d.) While you were in line to pay for your sandwich and drink, you just remembered that you hadn't purchased gifts for your 5 year old twin nephews' birthday party later in the afternoon. Not having time to purchase a gift, you decide to give each twin $10 in cash. How will you handle this transaction? You're now having one of those "Aha Moments" and know exactly how to masterfully receive the perfect amount of change, to make life simple for Sondra, to decrease the amount of coins in your pocket, and to have enough cash for your birthday gifts.
* Give Sondra your $20 and (2) $1 in addition to (1) Quarter, (1) Dime, and (1) Penny, which is a total of $22.36.
* Sondra is impressed with your astute Mathematics computation, and she returns a $10 and a $5 bill to you!
You win! Congratulations! You have successfully decreased the amount of coins in your pocket, and have made life easy for Sondra, and were able to place $10 in each birthday card for your twin nephews!
In my adult life, I have been instructed by a cashier from time to time that it was too late to make an adjustment to the amount that he/she had expected to receive from me, primarily in the form of a whole dollar bill, perhaps a $5, $10, or $20. I have been told by cashiers also, that they have been advised to provide the exact amount of change that the 'computer told them to give' and to not deviate from it. The trick and very first step is being confident and ready and to not show the cashier the amount you are paying until he/she is ready to receive your payment.
Let's take a look at a real-world example and see how you can adjust the amount of change you receive.
You are at Dave's Deli and have purchased a turkey sandwich and a large drink. The total due is $7.36, and you are paying cash for your purchase. Let's assume you have a $20 bill, a $10 bill, and (3) $1 bills, and the following coins: (3) Quarters, (1) Dime, and (4) Pennies. There is more than one solution to the amount that you may give Sondra, the deli's ace cashier.
Knowing that you don't have the exact change to give, here are some possible scenarios:
a.) Simply give her a $10 bill, forgetting that you have several coins in your pocket, and expect to receive $2.64 in change from Sondra.
b.) You remember that in addition to the $20, $10 and few $1 bills, you have several coins in your pocket. You quickly decide to pay her $10.36 and she returns $3 in one dollar bills.
c.) Sondra sees that you have some pennies in your hand, and she asks you to give her one penny in addition to the $10, making the total amount received $10.01. You are thinking to yourself, "Why is she asking me for an extra penny, when $10 is more than enough to cover your sandwich and drink?"
Sondra, seeing the confused look on your face, quickly explains that instead of returning $2 and $0.64 in coins, by giving her the extra penny, she now owes you change of $2.65 which simplifies the Math, saves her 4 pennies for the next customer, and you don't have to carry around extra coins in your pocket.
d.) While you were in line to pay for your sandwich and drink, you just remembered that you hadn't purchased gifts for your 5 year old twin nephews' birthday party later in the afternoon. Not having time to purchase a gift, you decide to give each twin $10 in cash. How will you handle this transaction? You're now having one of those "Aha Moments" and know exactly how to masterfully receive the perfect amount of change, to make life simple for Sondra, to decrease the amount of coins in your pocket, and to have enough cash for your birthday gifts.
* Give Sondra your $20 and (2) $1 in addition to (1) Quarter, (1) Dime, and (1) Penny, which is a total of $22.36.
* Sondra is impressed with your astute Mathematics computation, and she returns a $10 and a $5 bill to you!
You win! Congratulations! You have successfully decreased the amount of coins in your pocket, and have made life easy for Sondra, and were able to place $10 in each birthday card for your twin nephews!
Thursday, May 26, 2016
A Little Trick Using the Associative Property of Multiplication
A Little Trick Using Associative Property of Multiplication
Earlier this week, we reviewed the Associative Property of Addition. The Associative and Commutative Properties were the two properties that I had the most difficult time sorting out, for some reason.
As I teach my students each day, I encourage them to find a special way to memorize certain principles and formulas.
Let's take a closer look at the Associative Property of Multiplication. For many of us, we have many activities throughout life and we like to associate with certain friends as we enjoy those activities. There are those other activities that we enjoy, but we prefer to be with other groups of friends.
The Associative Property works very similarly to our social needs, and let's see how we can apply numbers to gain a better understanding. This property very simply states that if we are multiplying a few numbers, we can multiply them in groups and it doesn't matter which order we place them, the end result will be the same.
Here's a quick example:
1.) What is the value of 2 x (3 x 5)?
Solution:
Step 1--Do the parenthesis first, (3 x 5) = 15.
Step 2--Multiply by 2.
Answer: 2 x (3 x 5) = 30.
Now, let's regroup the numbers, using the parentheses around the first two numbers and see if we gain the same result.
2.) What is the value of (2 x 3) x 5?
Solution:
Step 1--Do the parenthesis first, (2 x 3) = 6.
Step 2--Multiply by 5.
Answer: (2 x 3) x 5 = 30.
We can easily create and demonstrate several problems, and the principle remains the same. As long as we are multiplying all of the numbers, we can reposition and regroup the parenthesis, and the result will be the same!
Create a few examples on your own, and you'll see how easy it is! Don't forget to share this great information with a friend!
As I teach my students each day, I encourage them to find a special way to memorize certain principles and formulas.
Let's take a closer look at the Associative Property of Multiplication. For many of us, we have many activities throughout life and we like to associate with certain friends as we enjoy those activities. There are those other activities that we enjoy, but we prefer to be with other groups of friends.
The Associative Property works very similarly to our social needs, and let's see how we can apply numbers to gain a better understanding. This property very simply states that if we are multiplying a few numbers, we can multiply them in groups and it doesn't matter which order we place them, the end result will be the same.
Here's a quick example:
1.) What is the value of 2 x (3 x 5)?
Solution:
Step 1--Do the parenthesis first, (3 x 5) = 15.
Step 2--Multiply by 2.
Answer: 2 x (3 x 5) = 30.
Now, let's regroup the numbers, using the parentheses around the first two numbers and see if we gain the same result.
2.) What is the value of (2 x 3) x 5?
Solution:
Step 1--Do the parenthesis first, (2 x 3) = 6.
Step 2--Multiply by 5.
Answer: (2 x 3) x 5 = 30.
We can easily create and demonstrate several problems, and the principle remains the same. As long as we are multiplying all of the numbers, we can reposition and regroup the parenthesis, and the result will be the same!
Create a few examples on your own, and you'll see how easy it is! Don't forget to share this great information with a friend!
Monday, May 23, 2016
A Little Trick Using Associative Property of Addition
I remember many years ago trying to memorize the handful of Math Properties, and some of them were easier to learn than others. The Associative and Commutative Properties were the two properties that I had the most difficult time sorting out, for some reason.
As I teach my students each day, I encourage them to find a special way to memorize certain principles and formulas.
Let's take a closer look at the Associative Property of Addition. For many of us, we have many activities throughout life and we like to associate with certain friends as we enjoy those activities. There are those other activities that we enjoy, but we prefer to be with other groups of friends.
The Associative Property works very similarly to our social needs, and let's see how we can apply numbers to gain a better understanding. This property very simply states that if we are adding a few numbers, we can add them in groups and it doesn't matter which order we place them, the end result will be the same.
Here's a quick example:
1.) What is the value of 2 + (3 + 5)?
Solution:
Step 1--Do the parenthesis first, (3 + 5) = 8.
Step 2--Add 2.
Answer: 2 + 8 = 10.
Now, let's regroup the numbers, using the parentheses around the first two numbers and see if we gain the same result.
2.) What is the value of (2 + 3) + 5?
Solution:
Step 1--Do the parenthesis first, (2 + 3) = 5.
Step 2--Add 5.
Answer: 5 + 5 = 10.
We can demonstrate several problems, and the principle remains the same. As long as we are adding all of the numbers, we can reposition the parenthesis, and the result will be the same!
Create a few examples on your own, and you'll see how easy it is! Don't forget to share this great information with a friend!
As I teach my students each day, I encourage them to find a special way to memorize certain principles and formulas.
Let's take a closer look at the Associative Property of Addition. For many of us, we have many activities throughout life and we like to associate with certain friends as we enjoy those activities. There are those other activities that we enjoy, but we prefer to be with other groups of friends.
The Associative Property works very similarly to our social needs, and let's see how we can apply numbers to gain a better understanding. This property very simply states that if we are adding a few numbers, we can add them in groups and it doesn't matter which order we place them, the end result will be the same.
Here's a quick example:
1.) What is the value of 2 + (3 + 5)?
Solution:
Step 1--Do the parenthesis first, (3 + 5) = 8.
Step 2--Add 2.
Answer: 2 + 8 = 10.
Now, let's regroup the numbers, using the parentheses around the first two numbers and see if we gain the same result.
2.) What is the value of (2 + 3) + 5?
Solution:
Step 1--Do the parenthesis first, (2 + 3) = 5.
Step 2--Add 5.
Answer: 5 + 5 = 10.
We can demonstrate several problems, and the principle remains the same. As long as we are adding all of the numbers, we can reposition the parenthesis, and the result will be the same!
Create a few examples on your own, and you'll see how easy it is! Don't forget to share this great information with a friend!
Sunday, May 22, 2016
Which Weighs More? Hint: It's Not the Ton of Bricks! 🤓
Many years ago, a teacher asked us which one weighed more, a ton of bricks or a ton of feathers. The natural response for most of us was to pick the what seemed like the obvious answer, which was a ton of bricks. Zooming right past the word 'ton' and focusing on the bricks, we knew, in our finite way of thinking, there was no possible way the feathers could weigh as much, or could they?
Thankfully, we were not on a TV Game Show, and didn't forfeit a life-changing Grand Prize by choosing the wrong answer!
As I reflect on this question and many others throughout my life, it often becomes a great teaching moment, as I once again consider the value of instructing others to think before we speak too quickly.
We know that a ton is a ton, whether we are talking about watermelons, gravel, bricks, or feathers. The correct answer to this riddle is they weigh the same.
Share this riddle and others with your children to sharpen their ability to think for themselves, and to be courageous and creative in their problem-solving skills.
Have a great day, everyone!
😀😀😀😀😀😀😀😀😀
Thankfully, we were not on a TV Game Show, and didn't forfeit a life-changing Grand Prize by choosing the wrong answer!
As I reflect on this question and many others throughout my life, it often becomes a great teaching moment, as I once again consider the value of instructing others to think before we speak too quickly.
We know that a ton is a ton, whether we are talking about watermelons, gravel, bricks, or feathers. The correct answer to this riddle is they weigh the same.
Share this riddle and others with your children to sharpen their ability to think for themselves, and to be courageous and creative in their problem-solving skills.
Have a great day, everyone!
😀😀😀😀😀😀😀😀😀
Saturday, May 21, 2016
Is it divisible by 5?
Sometimes you need to know if a number is evenly divisible by another number. Today's hint is quite simple when determining if a number is divisible by 5. If the dividend (the number being divided) ends with a 0 or 5, then it is always evenly divisible by 5.
Let's look at some examples.
1.) Is 164 evenly divisible by 5?
Solution:
Step 1--Does the number being divided (164) end with a 0 or 5? No!
Step 2--It is not evenly divisible by 5. You can certainly divide it, but you will have a remainder of 4.
The answer is 32.8 or 32 4/5.
2.) Is 90 evenly divisible by 5?
Solution:
Step 1--Does it 90 end with a 0 or 5? Yes.
Step 2--It is evenly divisible by 5. For every 100, 5 is divisible 20 times. Since 90 is 10 less than 100, you can conclude that not only is it evenly divisible, but the quotient (the answer to a division problem) is 2 less than 20, which is 18.
3.) Is 10,495 evenly divisible by 5?
Step 1--Does 10,395 end with a 0 or 5? Yes.
Step 2--It is evenly divisible by 5. For every 1,000 being divided, 5 is divisible 200 times. For 10,000 you can conclude that 10,000 is 10 x 1,000 and the answer to the first part of the solution is 10 x 200 = 2,000.
Step 3--Because we know that we can divide 5 into each 100 exactly 20 times, and 395 is 5 less than 400, we can conclude that 395 divided by 5 is simply 20 x 4 -1 = 79.
Step 4--Add 2,000 + 79 = 2,079.
Today's tip is easy to understand, and very practical if you need to divide a number by 5 without a remainder. Create and practice a few examples on your own and share this helpful tip with a friend! 😄
Let's look at some examples.
1.) Is 164 evenly divisible by 5?
Solution:
Step 1--Does the number being divided (164) end with a 0 or 5? No!
Step 2--It is not evenly divisible by 5. You can certainly divide it, but you will have a remainder of 4.
The answer is 32.8 or 32 4/5.
2.) Is 90 evenly divisible by 5?
Solution:
Step 1--Does it 90 end with a 0 or 5? Yes.
Step 2--It is evenly divisible by 5. For every 100, 5 is divisible 20 times. Since 90 is 10 less than 100, you can conclude that not only is it evenly divisible, but the quotient (the answer to a division problem) is 2 less than 20, which is 18.
3.) Is 10,495 evenly divisible by 5?
Step 1--Does 10,395 end with a 0 or 5? Yes.
Step 2--It is evenly divisible by 5. For every 1,000 being divided, 5 is divisible 200 times. For 10,000 you can conclude that 10,000 is 10 x 1,000 and the answer to the first part of the solution is 10 x 200 = 2,000.
Step 3--Because we know that we can divide 5 into each 100 exactly 20 times, and 395 is 5 less than 400, we can conclude that 395 divided by 5 is simply 20 x 4 -1 = 79.
Step 4--Add 2,000 + 79 = 2,079.
Today's tip is easy to understand, and very practical if you need to divide a number by 5 without a remainder. Create and practice a few examples on your own and share this helpful tip with a friend! 😄
Saturday, May 14, 2016
A Late-Night Laugh! 😂
I've often heard mathematicians proclaim that there are only 10 types of people in this world--those who understand binary, and those who don't!
😂😜😁😃😀😉😇😂
😂😜😁😃😀😉😇😂
Awesome Trick with Percents!
As a child, someone told me that those who are quick with their basic math skills, make successful negotiators. Whether we realize it or not, we negotiate deals many times throughout our lives. As I have experienced life, I realize more now than ever before, those words are great advice!
Today's tip relates to learning how to make quick mental calculations of percents that might otherwise require a calculator.
Take a look at this great information!
Suppose you need to calculate 12% of 50. Of course you can use the decimal form of 12%, which is 0.12, then multiply times 50. Doing the math, we know the answer is 6.
Here's an alternative way for you to calculate the product of these two numbers much quicker:
Step 1--Instead of calculating 12% of 50, switch the order of the numbers and trade the symbols, also.
Step 2--Calculate 50% of 12. We all know that 50% of any number is the same as dividing that number by 2. Very quickly, we know that 50% of 12 is 6. Same answer, but I'll bet you calculated it much quicker and with great ease!
Consider this real-world example:
Francine's Fashions is celebrating their 48th Anniversary this week, and has reduced all items in the store by 48%. (To find the savings, you need to know how to find 48% of $25.)
Step 1--Switch the numbers and the signs. Re-word the question to read, "What is 25% of 48?"
Step 2--Calculate 25% of 48. We know that 25% of any number is 1/4 of that number. We also know that we can multiply 48 x 1/4 to get the answer or divide 48/4.
Either way, you can quickly find the answer question without a calculator or paper and pencil! The answer is 12.
Try it, you'll quickly trade this method for the one you have been using! Don't forget to share this tip with a family member or friend! 😃
Today's tip relates to learning how to make quick mental calculations of percents that might otherwise require a calculator.
Take a look at this great information!
Suppose you need to calculate 12% of 50. Of course you can use the decimal form of 12%, which is 0.12, then multiply times 50. Doing the math, we know the answer is 6.
Here's an alternative way for you to calculate the product of these two numbers much quicker:
Step 1--Instead of calculating 12% of 50, switch the order of the numbers and trade the symbols, also.
Step 2--Calculate 50% of 12. We all know that 50% of any number is the same as dividing that number by 2. Very quickly, we know that 50% of 12 is 6. Same answer, but I'll bet you calculated it much quicker and with great ease!
Consider this real-world example:
Francine's Fashions is celebrating their 48th Anniversary this week, and has reduced all items in the store by 48%. (To find the savings, you need to know how to find 48% of $25.)
Step 1--Switch the numbers and the signs. Re-word the question to read, "What is 25% of 48?"
Step 2--Calculate 25% of 48. We know that 25% of any number is 1/4 of that number. We also know that we can multiply 48 x 1/4 to get the answer or divide 48/4.
Either way, you can quickly find the answer question without a calculator or paper and pencil! The answer is 12.
Try it, you'll quickly trade this method for the one you have been using! Don't forget to share this tip with a family member or friend! 😃
Thursday, May 12, 2016
Friday Funny--Gone Without a Tres!
The other day, I watched a magician who was famous for his amazing skills. He was very confident that he could do something very incredible that night, and he insisted that he could make himself disappear before the count of three, right in front of the audience.
Without hesitation, he began his routine.
"Uno," he said. Nothing happened.
"Dos," he counted, and poof! He was gone!
That's right, he disappeared without a tres!
😂😂😂😂😂😂😂😂😂😂😂😂😂😂
Without hesitation, he began his routine.
"Uno," he said. Nothing happened.
"Dos," he counted, and poof! He was gone!
That's right, he disappeared without a tres!
😂😂😂😂😂😂😂😂😂😂😂😂😂😂
Saturday, April 23, 2016
Math jokes--you should never have to explain them!
Have you ever told a joke, only to receive the response that someone doesn't understand the punchline? After several embarrassing situations where I have cracked jokes that were truly hilarious, and the response was just the opposite, I have learned to refrain from cracking intellectual jokes until I know the academic understanding of my audience.
When the audience has an appropriate understanding of academic humor, the response you typically receive from them is a joke that they've been dying to share with someone, but just could find the audience to listen.
Go ahead, crack some jokes! Good, clean, intellectual jokes are the best! Take time to laugh each day! 😄😂😄😂
When the audience has an appropriate understanding of academic humor, the response you typically receive from them is a joke that they've been dying to share with someone, but just could find the audience to listen.
Go ahead, crack some jokes! Good, clean, intellectual jokes are the best! Take time to laugh each day! 😄😂😄😂
Multiplying by 9?
As a young student, I was required to learn the multiplication table beginning with 0 all the way to 12 x 12, which is 144. As I teach my students each day, I am aware that there are some numbers in the multiplication table that are more difficult for them to learn than others.
Let's take a look at the number 9, and learn a basic trick.
9 x 1 = 09
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
9 x 6 = 54
9 x 7 = 63
9 x 8. = 72
9 x 9 = 81
9 x 10 = 90
Notice in the first column, containing all of the numbers highlighted in yellow, we begin counting with 0 and end with 9. In the second column, highlighted in mauve, we begin with 9 and count down to 0.
Although the 0 in front of the 9 is absent mathematically in the first calculation of 9 x 1, I have included it for illustrative purposes.
That's the trick! If you know someone who has difficulty mastering the multiplication of 9, please show them this practical tip!
Let's take a look at the number 9, and learn a basic trick.
9 x 1 = 09
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
9 x 6 = 54
9 x 7 = 63
9 x 8. = 72
9 x 9 = 81
9 x 10 = 90
Notice in the first column, containing all of the numbers highlighted in yellow, we begin counting with 0 and end with 9. In the second column, highlighted in mauve, we begin with 9 and count down to 0.
Although the 0 in front of the 9 is absent mathematically in the first calculation of 9 x 1, I have included it for illustrative purposes.
That's the trick! If you know someone who has difficulty mastering the multiplication of 9, please show them this practical tip!
Friday, April 22, 2016
Doggone decimal getting you down?
Have you ever needed to multiply two numbers and one of them contained a decimal? Without a calculator in hand, what was your immediate response? Did you panic? 😬
I have great news! 😀 There is hope for everyone! Know that our number system is based on a power of 10, and learn a few quick tricks, and you'll never panic again! 👍🏼
Let's take a look at an example, and learn about this nifty trick.
There are 10 children in the Schmidt family, and they have just received news that each of them will receive 1/10 of their grandmother's estate worth $445,000. What is the value of each child's inheritance?
Solution:
Step 1: Convert 1/10 to a decimal, which is 0.1
Step 2: Multiply $445,000 x 1. (That is not a typo. Multiply it times 1.)
Step 3: Apply the decimal to your calculation. You need to move the place value 1 position to the left only 1 place, since we are multiplying times 0.1.
The answer is $445,000 x 0.1 which is $44,500.
Each child will receive $44,500.
We'll look at a few more examples in upcoming posts. Don't forget to share this tip with a friend! 👌🏼
We'll look at a few more examples in upcoming posts. Don't forget to share this tip with a friend! 👌🏼
Thursday, April 21, 2016
Here's an average way of doin' it!
Have you ever been asked to find the average of two numbers, and wanted to know a simple way to find the solution without a calculator or paper and pencil? There is a very easy remedy, and most of us probably never realized how basic this process is.
Let's look at an example or two, and even use a real-world situation to illustrate.
John went bowling last night and wanted to know the average score of his first two games. He bowled a 242 in his first game, and earned a better score in his 2nd game, tallying an impressive 278! What was John's average so far?
Solution:
Step 1: Find the difference between the scores of the two games. 278 - 242 = 36.
Step 2: Divide 36 by 2, which is 18.
Step 3: Add 18 to the smaller number, which is 242 + 18 = 260.
All set! The average of 242 and 278 is 260.
Let's try another one.
Freddy cut grass for 2 of his neighbors today, and was paid $15 and $19. He wants to know the average earnings per lawn he mows, so he can estimate how many more he needs to cut to save enough money for his new bicycle.
Solution:
Step 1: Find the difference between $15 and $19, which is 4.
Step 2: Divide 4 by 2 = 2
Step 3: Add 2 to the smaller number, which is simply $15 + $2 = $17.
We have solved it rather quickly! The average of $15 and $19 = $17.
Additional Information:
To find the average of two numbers, you are merely finding the midpoint between those two numbers.
The midpoint between 7 and 9 is 8.
Look how easy this method is! Create a few more scenarios of your own, and don't forget to share this practical tip with a friend.
Let's look at an example or two, and even use a real-world situation to illustrate.
John went bowling last night and wanted to know the average score of his first two games. He bowled a 242 in his first game, and earned a better score in his 2nd game, tallying an impressive 278! What was John's average so far?
Solution:
Step 1: Find the difference between the scores of the two games. 278 - 242 = 36.
Step 2: Divide 36 by 2, which is 18.
Step 3: Add 18 to the smaller number, which is 242 + 18 = 260.
All set! The average of 242 and 278 is 260.
Let's try another one.
Freddy cut grass for 2 of his neighbors today, and was paid $15 and $19. He wants to know the average earnings per lawn he mows, so he can estimate how many more he needs to cut to save enough money for his new bicycle.
Solution:
Step 1: Find the difference between $15 and $19, which is 4.
Step 2: Divide 4 by 2 = 2
Step 3: Add 2 to the smaller number, which is simply $15 + $2 = $17.
We have solved it rather quickly! The average of $15 and $19 = $17.
Additional Information:
To find the average of two numbers, you are merely finding the midpoint between those two numbers.
The midpoint between 7 and 9 is 8.
Look how easy this method is! Create a few more scenarios of your own, and don't forget to share this practical tip with a friend.
Wednesday, April 20, 2016
Cut one number in 1/2 and double the other for easy multiplication!
Have you wondered if there is a trick to multiplying 2 numbers, especially if one number is much larger than the other? There is a neat trick that you can use every time, especially if one of the numbers has a factor of 2 multiplied several times. The only skill you need to know is how to cut one number in 1/2 and double the other number. You may need to repeat this step several times.
Let's take a quick look at a possible scenario.
Multiply 24 x 125. No need for paper and pencil or a calculator on this one! Check it out!
Step 1: Reduce 24 to 12 and Double 125 to 250.
Step 2: Reduce 12 to 6 and Double 250 to 500.
Step 3: Reduce 6 to 3 and Double 500 to 1,000.
We all know that 3 x 1,000 = 3,000. That's correct! 3 x 1,000 = 24 x 125
Here's another one!
After a great month at the sales office, Sales Manager, Ronald agreed to pay each of his 16 employees a $750 bonus! Let's see how easy this one will compute.
Step 1: Reduce 16 to 8 and Double $750 to $1,500.
Step 2: Reduce 8 to 4 and Double $1,500 to $3,000.
Step 3: Reduce 4 to 2 and Double $3,000 to $6,000.
Multiply 2 x $6,000 = $12,000. That's right! Ronald owes his employees $12,000 and they are so excited!
Tuesday, April 19, 2016
Multiply or Divide? Your Choice!
Is it easier for you to multiply or divide? Your answer to that question may change depending on the facts in the problem you need to solve.
Let's take a look at a couple scenarios, and you select which math operation you prefer to use.
1.) Sharon picked 48 ears of corn from her garden and wants to make enough equal portions so she will have 3 meals for her family. How many ears of corn will she be able to serve for each meal?
Option 1:
Solve by Division: 48/3 = 16.
Option 2:
Solve by Multiplication: 48 x 1/3 = 16.
* To solve, we multiplied by the reciprocal of 3, which is 1/3, and yields the same results as dividing by 3.
2.) Leonardo pumped 64 gallons of water into his new swimming pool, which took 4 minutes. How many gallons of water per minute did he pump?
Option 1:
Solve by Division: 64/4 = 16.
Option 2:
Solve by Multiplication: 64 x 1/4 = 16.
* Once again, we multiplied by the reciprocal instead of dividing. The reciprocal of 4 is 1/4, and as you can see, it yielded the same results. Leonard pumped 16 gallons of water into his pool each minute.
Create and solve a few examples on your own, and share this great tip with a friend! 😀
Let's take a look at a couple scenarios, and you select which math operation you prefer to use.
1.) Sharon picked 48 ears of corn from her garden and wants to make enough equal portions so she will have 3 meals for her family. How many ears of corn will she be able to serve for each meal?
Option 1:
Solve by Division: 48/3 = 16.
Option 2:
Solve by Multiplication: 48 x 1/3 = 16.
* To solve, we multiplied by the reciprocal of 3, which is 1/3, and yields the same results as dividing by 3.
2.) Leonardo pumped 64 gallons of water into his new swimming pool, which took 4 minutes. How many gallons of water per minute did he pump?
Option 1:
Solve by Division: 64/4 = 16.
Option 2:
Solve by Multiplication: 64 x 1/4 = 16.
* Once again, we multiplied by the reciprocal instead of dividing. The reciprocal of 4 is 1/4, and as you can see, it yielded the same results. Leonard pumped 16 gallons of water into his pool each minute.
Create and solve a few examples on your own, and share this great tip with a friend! 😀
Monday, April 18, 2016
Reciprocals are everywhere! How do you find them?
A reciprocal is indeed very valuable in many math calculations! Before we can properly use it, however, we must understand what it is.
A reciprocal is simply 1 divided that number. For example, the reciprocal of 2 is 1 divided by 2, or 1/2. The reciprocal of 4 is 1/4. The reciprocal of 25 is 1/25. Notice how dividing by 1 to find the reciprocal actually flips that number, so it is now on the bottom.
Negative numbers also have reciprocals and are calculated the same way. For example, the reciprocal of -6 = -1/6. The reciprocal of -9 = -1/9.
Every number has a reciprocal except 0. We know that it is not possible to divide any number by 0. 1/0 is undefined.
In our next post, we will see how reciprocals can be used to simplify our calculations! 😀
Friday, April 15, 2016
Need a pinch? A smidgen? A dash?
What in the world is a pinch? A dash? How about a smidgen? All of these measurements pertain to very small amounts of a fine powder or granular substance. Dry and liquid substances can vary, but will never differ by more than double, depending on the substance.
Here are the dry measurements of each:
Smidgen = 1/32 teaspoon
Pinch = 1/16 teaspoon
Dash = 1/8 teaspoon
Tad = 1/4 teaspoon
It is understood that the universal measurement for a pinch is simply the amount of a substance that you can pinch between your index finger and your thumb. Although some people argue that a pinch is not an exact measurement, it has been accepted that the amount of a pinch is equivalent to1/16 teaspoon.
Over the years, I've heard people request a smidgen when ordering a slice of cake or pie. If they only knew that a smidgen is 1/32nd of a teaspoon, that is hardly enough to taste! Why bother?! 😂
In some cookbooks, I've seen recipes listing a dash of salt or other seasoning. Doing some quick math, we can conclude that a dash, 1/8 teaspoon, is equivalent to 2 pinches.
A tad, which is 1/4 teaspoon, would equate to approximately 2 dashes or 4 pinches. When serving guests at my home, I've heard them ask for a tad of mashed potatoes and gravy. If they had known that one tad = 1/4 teaspoon, they would likely request a dollop, which is most definitely a more reasonable portion to eat! A dollop is a scoop or other mass that has no regular shape or form. I'll take a dollop of ice cream any day!
Would you like your own set of measuring spoons? You may order them at the link below.
If you are a regular Anazon customer, please save this link for your other purchases as well.
All prices and privileges are the same as other Amazon links.
www.astore.amazon.com/iwanttolovema-20
I hope you enjoyed this great tip! Remember, if you come to my home for dinner, I'll gladly serve you more than a smidgen of ice cream for dessert! I'll certainly top it with more than a tad of chocolate syrup!
Here are the dry measurements of each:
Smidgen = 1/32 teaspoon
Pinch = 1/16 teaspoon
Dash = 1/8 teaspoon
Tad = 1/4 teaspoon
It is understood that the universal measurement for a pinch is simply the amount of a substance that you can pinch between your index finger and your thumb. Although some people argue that a pinch is not an exact measurement, it has been accepted that the amount of a pinch is equivalent to1/16 teaspoon.
Over the years, I've heard people request a smidgen when ordering a slice of cake or pie. If they only knew that a smidgen is 1/32nd of a teaspoon, that is hardly enough to taste! Why bother?! 😂
In some cookbooks, I've seen recipes listing a dash of salt or other seasoning. Doing some quick math, we can conclude that a dash, 1/8 teaspoon, is equivalent to 2 pinches.
A tad, which is 1/4 teaspoon, would equate to approximately 2 dashes or 4 pinches. When serving guests at my home, I've heard them ask for a tad of mashed potatoes and gravy. If they had known that one tad = 1/4 teaspoon, they would likely request a dollop, which is most definitely a more reasonable portion to eat! A dollop is a scoop or other mass that has no regular shape or form. I'll take a dollop of ice cream any day!
Would you like your own set of measuring spoons? You may order them at the link below.
If you are a regular Anazon customer, please save this link for your other purchases as well.
All prices and privileges are the same as other Amazon links.
www.astore.amazon.com/iwanttolovema-20
I hope you enjoyed this great tip! Remember, if you come to my home for dinner, I'll gladly serve you more than a smidgen of ice cream for dessert! I'll certainly top it with more than a tad of chocolate syrup!
Thursday, April 14, 2016
Friday Funny! Swiss Flag
People sometimes ask me if I were to move to another country, which one would it be. As a Math Teacher, this is very easy to answer! Of course, I choose Switzerland! The delightful, scenic mountains and scrumptious chocolates are two good reasons, but take a look at that beautiful Swiss flag! It is definitely a big plus! 😄😄😄😄😄
Happy Friday, everyone!
Happy Friday, everyone!
Subscribe to:
Posts (Atom)