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Using Algebra to Do Arithmetic

You can certainly do basic arithmetic like 12 times 28 without algebra, but it’s interesting that algebra can be utilized to make some arithmetic calculations more efficient. This can be really handy when you don’t have a calculator around (or when you don’t want to waste your cell phone battery for math), or if you take a test where a calculator isn’t allowed.

As one example, consider the algebraic expression (x + 1)(x – 1). When you foil this out, you get x² + x – x – 1. The cross-terms vanish, leaving just x² – 1.

Knowing that (x + 1)(x – 1) = x² – 1 can actually be useful when doing multi-digit multiplication.

For example, consider 19 times 21. This is the same as (20 – 1)(20 + 1), which equals 20² – 1 or 399. It’s much easier to do 20 squared in your head than it is to work out 19 times 21.

Suppose you wish to multiply 26 by 34. You can write this as (30 – 4)(30 + 4) = 30² – 4² = 900 – 16 = 884.

Let’s try 13 times 17. You can turn this into (15 – 2)(15 + 2) = 15² – 2² = 225 – 4 = 221. Here, it helps to know the perfect squares of 11 thru 20.

The cross-term doesn’t always vanish, though. Consider 18 times 23. This becomes (20 – 2)(20 + 3) = 20² – 2(20) + 3(20) – 6 = 400 – 40 + 60 – 6 = 414.

As another example, 11 times 13 can be written as (10 + 1)(10 + 3) = 10² + 10 + 30 + 3 = 143.

When the cross-term doesn’t vanish, you’re probably not saving anything by using algebra – the usual arithmetic of 18 times 23 or 11 times 13 will be just as much work.

However, if you want to multiply 1001 times 1050, it may be simpler to write (1000 + 1)(1000 + 50) = 1000² + 1000 + 50000 + 50 = 1,051,050, provided that you’re good at counting the zeroes when you work with multiples of 10 and good at keeping track of decimal positions when you add (if not, the conventional method helps you stay organized).

Challenge yourself: Can you think of any arithmetic problems that would ordinarily be quite tedious, which algebra would make much simpler?

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

When Will Math Ever Be Useful?

Following are some examples of when various math skills may be useful in everyday life:

·         You’re buying several items in the store. If you can round, estimate, and do basic arithmetic, including percentages (to figure the tax), you should have an approximate total in mind when you approach the cash register. This way, you will know if you’re being overcharged because the employee made a mistake (or because the computer’s price didn’t match the labeled price).

·         You and two friends buy a large pepperoni pizza. One of your friends needs to leave as soon as the pizza arrives. He wants to take his share of the pizza with him. You need to be able to figure out what one-third of a pizza is in order to do this. You must understand fractions for this.

·         A department store is offering $10 off a purchase of $25 or more. You already found one item for $14.50. You want to buy a second item. You’re trying to figure out how much you need to spend in order to take advantage of the special offer. This is really an algebra problem at heart (solving for the unknown amount). However, you can figure this out if you can solve basic word problems with arithmetic.

·         One store has a sale where you buy three to get one free, while another store has a 30% off sale. Which sale is better? Understanding fractions and percentages can get you the better deal.

·         You want to buy several bricks to make a circular pattern in your backyard. How many bricks should you buy, given the diameter of the circle? Here, simple geometry can save you from buying too many or too few and having to make a second trip to the store.

·         You own a car that gets 30 miles per gallon on the highway. You want to make a roundtrip to a city that’s 1500 miles away. How much gas will you need to buy? How long will the trip take at 70 mph? This is a word problem, where understanding units (miles, gallons, and hours) can help; the math involved is just basic arithmetic.

 

You’re not going to carry a calculator with you everywhere you go. Sure, you can use your cell phone as a calculator, but doing so consumes valuable battery life.

 

And what happens when you type the problem incorrectly in your calculator? Students commonly get wrong answers on their calculators simply because they haven’t entered the information into it correctly. If you can round, estimate, and do basic math in your head, you can check if the calculator’s answer seems reasonable. This is a very valuable skill.

 

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

Free Periodic Table (Designed for Tests)

You may print and reproduce these periodic tables for non-commercial, educational purposes.

One periodic table is in full-color, while the other is black-and-white.

These periodic tables are designed for exam use, but may be used for other purposes, too:

• They include both the symbol and the name of the element, as chemistry instructors generally do not require students to memorize this information.
• The division between metals and nonmetals is shown with bold outline (but there are no labels for this – students are expected to know what this represents by the time they take an exam).
• They do not specify group names, electron structure, and other details that students may be expected to learn during a chemistry course.
• The older labeling of groups is used, as this aids students in deducing valence electrons for many elements. The new labeling of groups simply proceeds from left to right, so if you prefer the new labeling, you need only write the numbers in order at the top of the columns.
• Masses are given to the nearest 0.1 g, which is sufficient for most classroom instruction.
• Of course, atomic number is shown.

color periodic table

b&w periodic table

Chris McMullen, author of Understand Basic Chemistry Concepts and the Improve Your Math Fluency series of workbooks

The Power of One in Math

The number 1 is such a simple number:

·         It’s the first number when you count positive integers.

·         It has no effect when multiplying or dividing by 1.

·         It also has no effect as an exponent.

·         Adding or subtracting 1 simply yields the neighboring integer.

Yet the number 1 can be very powerful in math.

One way to convert units is to multiply by 1. For example, to convert 4 ft. to inches, you could multiply by 1, writing 1 as a fraction: 12 in. / 1 ft. equals 1. Multiplying 4 ft. by 12 in. / 1 ft. yields 48 in. (Observe that the feet cancel out.)

Fractions can be added by finding a common denominator, and the denominators are made common by multiplying by 1. For example, to add 1/4 to 1/3, the least common denominator is 12. Multiply 1/4 by 3/3 (which is 1) to make 3/12 and multiply 1/3 by 4/4 (also 1) to make 4/12. Now add these together to get 7/12.

The number 1 is used to turn numbers into their reciprocals. For example, the reciprocal of 5 is 1/5. The reciprocal of x is 1/x. Any number times its reciprocal equals 1. For example, 2 times 1/2 makes 1.

We use this principle all the time in algebra. For example, consider 4 x = 12. In this case, isolate the unknown, x, by multiplying by the reciprocal of its coefficient. The coefficient of x is 4; the reciprocal of 4 is 1/4. Multiply both sides by 1/4 to make x = 12/4 = 3. (Note that dividing both sides by 4 is equivalent to multiplying both sides by 1/4.)

The equation x + y⁰ = 2 wouldn’t have a unique solution if not for the fact that y⁰ = 1. (This follows from the rule that y^a/y^b = y^a–b. Let b = a. Then y^a/y^a = 1 = y^a–a = y⁰.) Knowing this, the original equation becomes x + 1 = 2, or x = 1.

In trigonometry, the number 1 often comes in handy through trig identities. For example, the equation 1 – sin²x = 0.5 cos x would be difficult to solve because it includes two different trig functions, except that 1 can be written as sin²x + cos²x, which reduces this equation to cos²x = 0.5 cos x. We can then factor it as cos x (cos x – 0.5) = 0, which means that cos x = 0 or cos x = 0.5, for which the solutions are 90, 60, or 300 degrees.

In probability, the number 1 represents 100%. If you reach into a bag with 3 red marbles and 5 blue marbles, you have a 3/8 probability of selecting red and 5/8 probability of selecting blue. If you add these together, you get 3/8 + 5/8 = 8/8 = 1. That’s because there is 100% that you’ll select a marble of some color.

There are many derivations and proofs in higher levels of math that become eloquent when a term is multiplied by 1 and the 1 is written in terms of an identity (such as the summation expression for the completeness theorem in group theory – which itself is another example of the power of one, in establishing unity for a complete set).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks

The Importance of Confidence in Mathematics – and How to Attain This

When learning mathematics, concepts often seem strange or intimidating to students. When this happens, many students proceed tentatively.

They sometimes give up when they were proceeding in the right direction; they just lacked the confidence to keep going.

They sometimes give up when they get the wrong answer, assuming that they solved the problem incorrectly, when all they did was make a simple mistake; they just lacked the confidence that the solution was correct to check it carefully. When you doubt your solution, you wonder if it’s worthwhile to check it over.

A student who lacks confidence in math is more likely to skip homework problems that seem too hard or rely on too much help from others. A more confident student will at least attempt the problems and is more likely to participate in the solution when receiving help. This small measure of confidence makes a world of difference.

The top students tend to be much more confident. They carry out their solutions fully because they’re sure they’re solving the problems correctly. Instead of doubting their solutions, when they get the wrong answer, they check their solutions carefully. Confidence plays a significant role.

The confident student is much more likely to find a solution to a challenging problem, whereas a tentative students is much more likely to give up easily.

Overconfidence can also be a problem. The overconfident student may not practice or study as much as necessary, or might not put enough thought and care into a solution.

How can students develop confidence in math?

It’s not easy for everyone to show confidence; some students have more trouble with this than others. But everyone can become confident.

It’s not just a matter of saying you’re confident. There’s much more to it than this.

Suppose you don’t know a word of Russian and suddenly wander down the streets of Moscow telling yourself you’re confident you can easily pick up on the language. Not gonna happen. (Maybe someone will speak English, but you’re not going to instantly pick up Russian no matter how confident you are.)

Students can become confident through experience.

The student who hasn’t practiced or studied enough definitely lacks confidence.

The top students know they have practiced plenty and studied hard, so they show confidence.

Then there are good students who worked and studied hard, but don’t show confidence; instead, they show much anxiety and make nervous mistakes. These students didn’t first convince themselves that because they’ve worked and studied so hard, they do know how to solve the problems. They had every reason to be confident. Maybe they have done poorly on tests in the past, and this prevented them from showing the needed confidence.

There is a way for students to overcome math exam anxiety: One step at a time. Try to solve one small problem by yourself. Then work your way up to exam conditions. Start with self-check exercises, try practice quizzes, make a practice exam. Put yourself in positions where you experience exam anxiety, where it starts out easy and becomes progressively more challenging. It takes a will to find the way.

Teachers or parents can help students with this through practice problems, practice quizzes, and practice tests. The idea is not to give a preview of the actual problems, but to recreate similar exam conditions in order to help students overcome their exam anxiety. After every practice test or actual test, spend time with the student discussing the student’s approach.

Chris McMullen, author of the Improve Your Math Fluency Series

Hem

Issues with Multiple Choice Tests in Algebra

Suppose that an algebra problem asks a student to solve an equation for an unknown. The goal of the problem is to test whether or not the student knows how to apply a particular algebraic technique.

However, if the question is multiple choice, the student doesn’t actually need to know how to solve the equation in order to determine the correct answer. The student could simply plug each answer into the equation to see which one works.

Example: 3x + 20 = 12x + 2.

(A) x = 1 (B) x = 2 (C) x = 3 (D) x = 4 (E) x = 5

A student could simply try each answer. Plugging in x = 1, it’s easy to see that 23 doesn’t equal 14. Trying x = 2, 26 = 26. The student already has the answer, but hasn’t done any algebra.

If the correct answer is (E), or if the student makes a mistake in the calculation, it will take longer, but in principle, a student can get the correct answer without doing any algebra.

If there are fractions in the answers, that makes the calculation a little more cumbersome. Many students prefer to avoid fractions, so this may deter students from avoiding the algebra.

Suppose that the choices had been:

(A) x = 1/2 (B) x = 18/15 (C) x = 22/15 (D) x = 2 (E) x = 22/9

Some problems can be solved faster by actually doing the algebra. But more time-consuming problems, like using the quadratic or solving a system of equations, might be solved faster by just plugging the answers into the original equations. When algebra yields the fastest solution, this provides an incentive to solve the problem as intended.

If a student relies on plug and chug for every problem, the student risks not finishing the test. But students probably won’t try to solve every problem this way. Just the hard ones.

Fortunately, there are many problems that tests can ask that can’t be solved this way. For example, many problems ask students to simplify an expression, and there are also conceptual, strategic, and logical questions. This helps to ensure that students must grasp some algebra concepts in order to pass the class.

The main goal of an algebra course is for students to become fluent in solving for unknowns in a variety of types of equations. Multiple choice is convenient, especially in large classes. It’s worth considering whether or not students might find ways to succeed in the course without actually mastering the techniques.

One way is for the instructor to become acquainted with the students, perhaps by checking written solutions of homework or quizzes (or adding a few written problems to exams, where feasible), helping students solve problems on the board, or interacting with students during office hours. If students who ordinarily struggle solving equations are better able to figure out the right answers on multiple choice problems, this might be a signal (of course, this could also be the result of studying, tutoring, and improvement). If instead experience with their written solutions corresponds well with their ability to solve similar problems on multiple choice exams, then there may not be any reason to worry that the students who most need to improve their fluency might be finding an easy way out (which may very well be the case for many of the students – it might be the clever problem-solvers who are most likely to think of this).

Chris McMullen, author of the Improve Your Math Fluency series of workbooks