Important Reasons for Learning the Fundamentals of Math
Important Reasons for Learning the Fundamentals of Math
October 28th, 2008 | by Sol |
[ Editor’s note: This post was contributed by Kelly Kilpatrick, who writes on the subject of how to Become a teacher in Florida. She invites your feedback at kellykilpatrick24 at gmail dot com. ]
Math is used in varying degrees in every single aspect of our daily lives, especially in the information age. Whether you are cognizant of this fact or not, computations are constantly running calculations are being made, and formulas are being proved. Our world is a mathematical world. What follows is a list five important reasons we need to have a strong understanding of the fundamentals of math.
Math is an integral part of daily life.
Whether you are balancing your checkbook or receiving change from the attendant at the gas station, math is being used. Mathematical processes are hard at work behind the scenes, keeping things working and operating smoothly. Having a firm understand of the basic principals of math is necessary to function and prosper on a daily basis.
Fundamentals are the building blocks for more advanced math.
If you have a firm grasp of the fundamentals of math, you will be able to learn more advanced mathematical processes far more easily. Having good math skills will ultimately save you time and reduce the need to need tutoring or remediation. Fundamentals are needed to move on; not having this foundation will result in more time spent on working on things that you should already know. Additionally, since each process builds upon prior knowledge and successful application of these skills, it is of the utmost importance that the fundamentals are solid.
Skills increase your speed and accuracy.
When time is of the essence regarding coursework or anything else that requires math skills, it is good to have your fundament math skills down pat. Your work will require less checking if you are certain that you have done the finer points correctly, and with accuracy comes speed naturally.
Early success is the beginning of future success.
Since our world is a mathematical world, it logically follows that establishing your simple math skills early on will ensure that you have the skills necessary to work toward success related to later mathematical endeavors. Whether you are interested in computer science, engineering, or any number of professions that utilize math on a regular basis, it behooves you to hone your mathematical skills sooner rather than later.
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This post was contributed by Kelly Kilpatrick, who writes on the subject of how to Become a teacher in Florida. She invites your feedback at kellykilpatrick24 at gmail dot com.
MULTIPLICATION IS FUN
1. Multiplying by 9, or 99, or 999
Multiplying by 9 is really multiplying by 10-1.
So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81.
Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414.
One more example: 68×9 = 680-68 = 612.
To multiply by 99, you multiply by 100-1.
So, 46×99 = 46x(100-1) = 4600-46 = 4554.
Multiplying by 999 is similar to multiplying by 9 and by 99.
38×999 = 38x(1000-1) = 38000-38 = 37962.
2. Multiplying by 11
To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.
Let me illustrate:
To multiply 436 by 11 go from right to left.
First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.
Write down 9 to the left of 6.
Then add 4 to 3 to get 7. Write down 7.
Then, write down the leftmost digit, 4.
So, 436×11 = is 4796.
Let’s do another example: 3254×11.
The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.
One more example, this one involving carrying: 4657×11.
Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).
Going from right to left we write down 7.
Then we notice that 5+7=12.
So we write down 2 and carry the 1.
6+5 = 11, plus the 1 we carried = 12.
So, we write down the 2 and carry the 1.
4+6 = 10, plus the 1 we carried = 11.
So, we write down the 1 and carry the 1.
To the leftmost digit, 4, we add the 1 we carried.
So, 4657×11 = 51227 .
3. Multiplying by 5, 25, or 125
Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.
12×5 = (12×10)/2 = 120/2 = 60.
Another example: 64×5 = 640/2 = 320.
And, 4286×5 = 42860/2 = 21430.
To multiply by 25 you multiply by 100 (just add two 0’s to the end of the number) then divide by 4, since 100 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4.
64×25 = 6400/4 = 3200/2 = 1600.
58×25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2×2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times.
32×125 = 32000/8 = 16000/4 = 8000/2 = 4000.
48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.
4. Multiplying together two numbers that differ by a small even number
This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.
Let’s say you want to calculate 12×14.
When two numbers differ by two their product is always the square of the number in between them minus 1.
12×14 = (13×13)-1 = 168.
16×18 = (17×17)-1 = 288.
99×101 = (100×100)-1 = 10000-1 = 9999
If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.
11×15 = (13×13)-4 = 169-4 = 165.
13×17 = (15×15)-4 = 225-4 = 221.
If the two numbers differ by 6 then their product is the square of their average minus 9.
12×18 = (15×15)-9 = 216.
17×23 = (20×20)-9 = 391.
5. Squaring 2-digit numbers that end in 5
If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.
35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.
To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer.
85×85: Calculate 8×9 = 72 and write down 7225.
6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10
Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.
An illustration is in order:
To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20.
Multiply together the last digits: 2×8 = 16. Write down 16.
The product of 42 and 48 is thus 2016.
Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.
Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224.
A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224
7. Squaring other 2-digit numbers
Let’s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40.
Double this product: 40×2=80, then add a 0 to it, getting 800.
Add 800 to 2564 to get 3364.
This is pretty complicated so let’s do more examples.
32×32. The first part of the answer comes from squaring 3 and 2.
3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.
Multiply the digits, 2 and 3, together and double the whole thing. 2×3×2 = 12.
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56×56. The partial product comes from 5×5 and 6×6. Write down 2536.
5×6×2 = 60. Add a zero to get 600.
56×56 = 2536+600 = 3136.
One more example: 67×67. Write down 3649 as the partial product.
6×7×2 = 42×2 = 84. Add a zero to get 840.
67×67=3649+840 = 4489.
8. Multiplying by doubling and halving
There are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.
Let’s say you want to multiply 14 by 16. You can do this:
14×16 = 28×8 = 56×4 = 112×2 = 224.
Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180.
48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.)
9. Multiplying by a power of 2
To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2×2×2.
15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240.
23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184.
54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432.
Practice these tricks and you’ll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you’ll get to choose which one is easiest for a particular problem.
Multiplication can be a great sport! Enjoy.
THE IMPORTANT OF MATHEMATICS
Importance of Mathematics for a Future Career
1. How important is mathematics as an entrance requirement to university programs?
Students who choose to ignore Mathematics, or not take it seriously in High School, forfeit many future career opportunities that they could have. They essentially turn their backs on more than half the job market (see the areas listed below). The vast majority of university degrees require Mathematics. The importance of Mathematics for potential future careers cannot be more emphasized.
For example, degrees in the following areas require good knowledge of Mathematics and Statistics:
- the physical sciences (like Chemistry, Physics, Engineering),
- the life and health sciences (like Biology, Psychology, Pharmacy, Nursing, Optometry),
- the social sciences (including Anthropology, Communications, Economics, Linguistics, education, Geography)
- the tech sciences (like Computer Science, Networking, Software development),
- Business and Commerce,
- Actuarial science (used by insurance companies)
- Medicine
2. That marks are important and have a strong bearing on, not only entrance to university, but also success upon their arrival in their first year.
Yes, students would be wise to enter university with a good background in Mathematics. Experience has shown that students who come to university with a poor grade in Math (or who choose to skip Math in high school) have a difficult time progressing in the disciplines they have chosen to major in. So we strongly recommend students take Mathematics seriously during their high school years and score at least a B to be able to do reasonably well in university Mathematics. This is in the student’s best interest. Nevertheless, the Mathematics Department at UNBC offers MATH 115 for students who did not take MATH 12, to help such students bridge their transition to university life.
3. The challenges that face first year students. (Work load, Math topics covered, how students cope with the transition from high school.)
Indeed, there are cultural differences and challenges. In the university setting students are treated as adults who are responsible for their own welfare and course work. Normally, students take 5 courses per semester so as to complete the degree requirements in 4 years (which are normally 120 credits, though some degrees have more). However, help is available in terms of tutorials, support from the Learning Skills Center, office hours, and collaboration with fellow students. Our faculty have a proven record of being helpful and supportive of their students (something which we often hear rumored by our students).
4. The many uses of mathematics in the university setting.
The answer to question 1 shows just how pervasive Mathematics is in the university and in the many other degrees that offers. Mathematics courses are required for students who want to major in the following areas: Business, Marketing, Finance, Commerce, Biology, Economics, Chemistry, Computer Science, Physics, Environmental Engineering, Forestry, Psychology, Nursing, Health and Human Sciences.
5. Your experiences with mathematics as it is used in a variety of career areas.
Every area of Mathematics has its own unique applications to the different career options. For example, Algebra is very important for computer science, cryptology, networking, study of symmetry in Chemistry and Physics. Calculus (including differential equations) is used in Chemistry, Biology, Physics, Engineering, the motion of water (hydrodynamics), rocket science, molecular structure, option price modeling in Business and Economics models, etc.
6. Any other motivational topic that will help students understand the competative nature of the working world and the importance of being successful in math.
Students are encouraged to give serious attention to their future. The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are excluding themselves from the many career paths that need Mathematics. We therefore exhort all our high school students to take matters in their own hands, to study hard, achieve a level of excellence, and take such a fundamental discipline like Mathematics seriously if they think they may go into any of the Physical, Social, Health Sciences, Business, Medicine, or related areas.
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