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