26

It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...


17

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 Fahrenheit when given the Celsius temperature: (1) Double the temperature in Celsius, (2) subtract off 10% (something that is easy to calculate), (3) add 32.


10

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 $46^2 = (40+6)^2 = 40^2+2\cdot40\cdot6 +6^2 = 1600 + 480 + 36 = 2116$. You can also use this method for negative $b$: $ 197^2 = (200 - 3)^2 = 200^2 - 2\cdot200\cdot3 + 3^2 = 40000 - 1200 +...


8

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 $b^2$ smaller than the square. So 45 squared is (40*50)+25 or 2025. And any multiplying where you can latch on to an easy head math where the true numbers are +/- the ...


7

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 wouldn't let me use it on the test! Example: $0.36298298298\overline{298} \ldots$ (0.36298 with the 298 repeating). Explanation: 1) take the complete part of ...


6

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 problem is to divide the six-bit number 110000 into the thirteen-bit number 1000100010000. Scanning the bits of the second number(the dividend) from left to ...


6

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 further: If you memorise the base 10 logarithms of 2, 3 and 7, you can quickly deduce the logs of all the other digits in your head and amaze people with ...


6

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. 4, if the second last digit is even, ask whether the last digit is 0, 4, or 8, otherwise ask whether it the last digit is 2, 6. 5, ask whether the last digit is ...


6

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 better than the form of long division I learned. "In fact a lot of kids find it [short division] amazing to learn." If you show an individual a new process at ...


5

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 these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. ...


5

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 requested by the OP, I think the value of the book is to give a taste for mental constructions for calculation that feel an awful lot like what mathematicians do ...


5

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 \\ 111\cdot111&=12321 \\ 1111\cdot1111&=1234321 \\ 11111\cdot11111&=123454321 \\ 111111\cdot111111&=12345654321 \\ 1111111\cdot1111111&=...


4

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 of this in particular, but many of the debates around "Common Core mathematics" include disagreements on whether it is beneficial for students to learn ...


4

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 "back of the envelope computations" (or even just getting rough estimates, so you aren't embarrased when the finance minister says that "well, 10% of 10% is ... ...


3

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 grade age). The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't ...


3

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 multiply and check, either one. Take $8$ for an example. $48\times8=384$ and $436-384=52$. By looking at what we have left, we know that we need only one more $48$ ...


2

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 left to right. With the usual algorithm for multiplying by hand where the answer is produced right to left, it looks like you've already computed the product when ...


2

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 $m$ (most people know the squares of the natural numbers up to 20), add the two and then add the result to 2500. To illustrate, let $n = 62$, then $m=12$ and $...


2

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 $$\left( a+\frac{1}{2}\right)^2 = a(a+1)+\frac{1}{4}$$ or $$\left( a+\frac{1}{2}\right)^2 = a^2+a+\frac{1}{4}$$ We can quickly see that $\sqrt{72}$ is between $8$ and $9$, ...


1

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 division algorithm is long. Remembering the steps can be difficult. The long division algorithm may not be intuitive. We can tell them that it's just "how many groups ...


1

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 square 1) Perform a multiplication is the easy one : In order to multiplicate two numbers, calculate their average and the difference to that average, and ...


1

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.


1

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 $x = 3$. $$P(3) = (3)^3-7(3)^2+6(3)-8$$ $$= 3(3^2)-7(3^2)+2(3^2)-8$$ $$= (3-7+2)(9)-8$$ $$= -2(9)-8 = -18-8=-26$$ It comes up more often than one might expect in textbook ...


1

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 \times 13 = 156$ from $100 + 50 + 6$ When we go higher than the products $12 \times 14$ or $13^2$ care is needed in carrying over, $13 \times 15 =100 + 80 + ...


1

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 published in 1965. It contains a list of mental calculation techniques claimed to be based on the Vedas. The mental calculation system mentioned in the book is also ...


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