"Borrow" vs. "Shift"
Most can remember subtraction using the classic "borrowing" method, such as,
63 - 28 = (50 + 13) - (20 + 8) = (50 - 20) + (13 - 8)* = 35.
Here we "borrowed" from the 60, adding 10 to the 3 for 13, and leaving 50 behind to subtract the 20 from. This all looks more familiar if one writes it out in the traditonal way, vertically, crossing out the 6, writing the 1 by the 3, etc. You know, "borrowing."
This is a way to do this, but not the only one. We also can change the problem a bit, and into one that gives the same answer. We do this by "shifting" the numbers an equal amount, in this case by adding 2 to both. Here it is.
63 - 28 = (63 + 2) - (28 + 2) = 65 - 30 = 35.
We’ll start with the easy stuff.
Multiplying by 4:
(1) Double the multiplicand you want to multiply 4 by
(2) Double it one more time
e.g. 8 * 4 = 32
8 * 2 = 16
16 * 2 = 32
Why does this work?
4 can be broken up into 2 * 2
8 * 4 = 32
8 * (2 * 2) = 32
Thanks to the associative property of multiplication, we can multiply factors in whatever grouping or order we chose and still get the same answer. We start by multiplying multiplicand we want to multiply 4 by 2 because this computation is easy for most people to do in their heads.
(8 * 2) * 2
(16) * 2
We then multiply our product by the remaining multiplicand, which is 2.
16 * 2 = 32
Multiplying by 10:
Stick a zero behind whatever number you wish to multiply 10 by
988 * 10
Why does this work?
Consider what we’re doing in terms of place value. When...
Welcome back to the school everyone! I hope you all had a great summer. For all those whose summer was maybe a little
too great, maybe those who’ve forgotten even the basics, we’re going to take it all the way back to arithmetic a.k.a “number theory”.
A review of number theory is a perfect place to start for many levels. Calculus and a lot of what you learn in pre-calculus is based on the real number system.
When we use the word “number” we are typically referring to all real number. But how can numbers be “real”? You can’t touch the number 6 or smell 1,063. You can’t boil 1/2 or stick it in a stew. So what’s so real about real numbers? The simple answer is this:
a real number is a point on a number line (1).
-2.5 -1 0 ...
Summary: Mental math teaches students to see short, efficient solutions—rather than to blindly follow
the brute-force, cookie-cutter, one-size-fit-all, show-all-your-work procedures taught at school.
To my youngest students, I lie—by omission—that vertical arithmetic does not exist. I can usually get away with it for about a year. Until the school shows them the light. Say, how to add 25 and 8
vertically, with the carry-over 1 carefully written on top of the 2. By that time, my students are proficient in mental addition and subtraction of 3-digit numbers: carrying, borrowing, and all. My goal though is by no means to turn them into human calculators. So then, why bother?
Vertical arithmetic is a convenient method for computing numerical answers. Especially when the numbers to manipulate are multidigit. But it is a
procedure, requiring—once learned—little thought. The entire process is delegated to the...
Dazzling pocket PCs are aplenty for the children of today. Kids roll into the classroom with iPhones, Blackberries, and various Android devices capable of supporting myriad complex applications. We are living in a wonderful age where handheld computers help us tremendously and continuously. Alongside all of the fancy apps (that allow us to manage everything from our finances to our fantasy football teams) is a standard utility application that accompanies every smartphone: the basic calculator. Need to carry out some quick arithmetic to figure out how much money you owe your buddy? Pull out your phone and type away. It’s that simple. So why the heck do kids need to memorize the multiplication table? Because it is still crucial to a successful math career and a promising life thereafter. Don’t believe me? Here are four reasons why mental math is still tremendously important and absolutely foundational.
1. Confidence Is Key
You have likely heard people utter the following...
In elementary school, mathematics is often taught as a set of rules for counting and computation. Students learn that there is only one right answer and that the teacher knows it. There is no room for judgment or making assumptions. Students are taught that Arithmetic is the way it is because it's the truth, plain and simple. Often students go on to become trapped in this view of the universe. As fairy tales fade from the imagination, so is mathematical creativity lost.
There is evidence that Mathematics and Arithmetic existed over 3000 years ago, but only the very well educated leisure class had access to it. The rules for simple computation only were developed recently, so much of the computation of sums and products was much more complicated. Imagine adding and multiplying Roman Numerals for example. Because of this difficulty, computations were laid out only to solve very specific practical problems.
Although mathematics was mainly limited to solving practical...
When working with fractions, I find it effective to require students to convert each fraction that we work with to its decimal equivalent, to convert that decimal equivalent back into the original fraction, to convert that decimal into its percentage equivalent, to work a simple percentage problem using that percentage and finally to work the same problem using the initial fraction.
This comprehensive method helps students to see the relationships between fractions, decimals and percentages in a holistic way and to promote the necessary skills in each element.