17 votes

Tricks for computing things in your head

I never divide or multiply by 5. I'll double and divide by 2 or divide by 10 and double. In a similar manner, I was in China last summer and found a quicker way to calculate the temperature in ...
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10 votes

Tricks for computing things in your head

Some tricks I've seen: Tricks with notable products $(a + b)^2 = a^2 + 2ab + b^2$ This formula can be used to compute squares. Say that we want to compute $46^2$. We use $46^2 = (40+6)^2 = 40^2+2\...
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  • 915
9 votes

Tricks for computing things in your head

I think $(a+b)(a-b) = a^2 - b^2$ is the most underutilized trick. A bit of practice and it's easy to see how the square is $b^2$ bigger than the rectangle, and conversely, the rectangle $b^2$ ...
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7 votes

Tricks for computing things in your head

I devised a trick in 8th grade for converting repeating decimals to fractions. They were teaching a very long drawn-out process. My trick basically does the same thing but for some reason they ...
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  • 219
6 votes
Accepted

Determining the first digit of the Quotient using hand long division efficiently?

The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1. In binary, your ...
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  • 7,575
6 votes

Tricks for computing things in your head

Because of an exercise we ran every year, I accidentally memorised $\log_{10}2=0.301$, and I could find many logarithms quickly in my head using the log laws. One holiday I decided to take it a little ...
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  • 1,912
6 votes

Tricks for computing things in your head

To tell if a number is divisible by 2, ask whether the last digit even. 3, if there is one digit, ask whether it is 0, 3, 6, or 9, otherwise add the digits and ask whether the sum is divisible by 3. ...
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6 votes

Is short division taught these days and if not, why not?

If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much ...
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  • 17k
5 votes

Is short division taught these days and if not, why not?

Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure ...
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5 votes

Tricks for computing things in your head

This book of Art Benjamin is fantastic. There is a new version of it, but I cannot remember the title. I should mention that although the book contains many specific "tricks for calculation" as ...
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  • 5,887
5 votes

Tricks for computing things in your head

The squares of the first couple of numbers using only the digit one show a nice pattern and are thus easy to remember. They are collected in the list below. $$\begin{align} 11\cdot11&=121 \\ ...
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4 votes
Accepted

What's the word for addition and subtraction without borrowing or carrying over?

This topic seems to be called "Bridging 10s" or "Making 10s": http://www.helpingwithmath.com/by_subject/addition/making-10-1oa6.htm I'm not sure that you will find many discussions about the utility ...
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  • 3,477
4 votes

Tricks for computing things in your head

I agree that the "this is the way this <cool trick> works" can be very motivating, but I see it much more important to learn to do (and use!) what Jon Bentley in his "Programming Pearls" calls "...
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  • 12k
3 votes

Is short division taught these days and if not, why not?

Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st ...
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  • 161
3 votes

Determining the first digit of the Quotient using hand long division efficiently?

Round. Check. Correct. What we need is $436\div48$. Rounding to remove the last digit reduces the problem to $44\div5$. This gives us an educated guess of either $8$ or $9$. From here we then ...
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2 votes

Tricks for computing things in your head

Quick "multiplication" Ask your students to call out digits until you have the entire width of the board filled. Then write $\times\,5$ on the next line. Proceed to right the full answer out from ...
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  • 121
2 votes

Tricks for computing things in your head

Squaring natural numbers around 50 Let n be an integer "around 50" (which can be anywhere from 30 to 70). If we write $n$ as $n = 50+m$ then $n^2= 2500+100m+m^2$. So just multiply $m$ by 100, square ...
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2 votes

Tricks for computing things in your head

Multiplying whole numbers times $25$ is counting quarters. Many American students find this easy. Getting a better that whole number approximation of square roots is possible using $$\left( a+\frac{...
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  • 351
1 vote
Accepted

Math dyslexia is a big problem for me. I lag behind my classmates

Have you tried bringing this to your math teacher's superiors? It seems to me a perfectly good case for them to make an exception to the "no calculator" rule. Normally teachers/schools don't ...
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1 vote

Is short division taught these days and if not, why not?

Doing a quick google search for "why do students struggle with division?" made me realize myself why long division is difficult for a lot of students. Common reasons claimed include: The long ...
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1 vote

Tricks for computing things in your head

Let me give you some insight how I calculate things by heart: First we start with the simple formula: $a^2-b^2=(a-b)*(a+b)$ This can be used for two things: perform a multiplication calculate a ...
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  • 832
1 vote

What's the word for addition and subtraction without borrowing or carrying over?

When I studied those things, in Spanish it's was called "suma algebraica", i. e., algebraic sum. Numbers in a term could be positive or negative.
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  • 161
1 vote

Tricks for computing things in your head

It's not really mental math, but it is more useful that one might expect. And it encourages students to see factoring and grouping possibilities. Evaluate $P(x) = x^3-7x^2+6x-8$ at $x = 3$. $$P(3) = ...
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  • 351
1 vote

Tricks for computing things in your head

To multiply two numbers from $11 - 19$, written in base ten as $1a$ and $1b$, we can use, $(10 + a)(10 + b) = 100 + (a + b)10 + ab$ It could be remembered as 'One add multiply'. For example, $ 12 ...
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  • 333
1 vote

Tricks for computing things in your head

This might be of interest to you - http://en.wikipedia.org/wiki/Vedic_Mathematics_(book) Vedic Mathematics is a book written by the high-ranking Hindu cleric Bharati Krishna Tirthaji and first ...
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